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© Fred E. Lusk III The Four Fours Game 04/24/1993~07/07/1998~01/25/2012 4S00_INTRO.DOCX THE FOUR FOURS GAME Fred E. Lusk III, P.E. January 25, 2012 Version 3.30

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THE FOUR FOURS GAMEFred E. Lusk III, P.E.January 25, 2012Version 3.30© Fred E. Lusk III 04/24/1993~07/07/1998~01/25/2012The Four Fours Game 4S00_INTRO.DOCXDEDICATIONThis booklet is dedicated to Mr. Sherard, Math Teacher Extraordinaire at the old Sierra Junior High School in Fresno, CA, who introduced me and my classmates to The Four Fours Game. This booklet is also dedicated to my father, who showed me the factorial function and a few others; to my mother, who helped with some of the

Text of The Four Fours Game

Fred E. Lusk III The Four Fours Game 04/24/1993~07/07/1998~01/25/2012 4S00_INTRO.DOCX Fred E. Lusk III, P.E. January 25, 2012 Version 3.30 Fred E. Lusk III The Four Fours Game 04/24/1993~07/07/1998~01/25/2012 4S00_INTRO.DOCX DEDICATION This booklet is dedicated to Mr. Sherard, Math Teacher Extraordinaire at the old Sierra Junior High School in Fresno, CA, who introduced me and my classmates to The Four Fours Game. This booklet is also dedicated to my father, who showed me the factorial function and a few others; to my mother, who helped with some of the harder solutions; to my friend Scott Endler, who competed with me for top honors in the game; and to my wife, three children, and now two grandchildren, who put up with all this nonsense. Any mistakes herein are mine alone, of course, though I am looking for someone else to blame. DOCUMENT HISTORY 1993: This treatise was originally prepared with Microsoft Word for Windows 2.0 and some of its goodies, including the Equation Editor and WordArt, operating under MS-DOS 5.0 and Windows 3.1 on a Standard Technologies 486-33 with 8MB RAM (which wasn't always enough, even with a really big swap file). 1998: The next installment was prepared with Microsoft Word for Windows 6.0 operating under Windows 95 on a Gateway 2000 P5-90 with 24 MB RAM (which wasn't always enough). The 1993 through 1998 work was made even more difficult because documents with lots of equations sometimes behaved very badly during editing. 2012: After a long hiatus, work continues with Microsoft Word 2010 operating under Windows 7 on a 3.0-GHz Dell Dimension 8400 with 2.0 GBytes of RAM (again, not always enough). Besides updating the narrative and adding more solutions, a key task was replacing all the old Equation Editor objects with new Equation Editor objects. Now the documents are happy unless I am editing equations with several documents open at once.. Copyright 1993, 1998, 2012 Fred E. Lusk III, P.E., Fresno, California Yall have my permission to make and distribute electronic and/or paper copies of The Four Fours Game subject to the following stipulations: 1. All pages must include my name, the copyright symbol and dates, and the filename. Theyre already there, so dont mess withem. If you have the PDF version, you wont be able to mess withem. 2. Electronic copies must include the following two files exactly as I publish them: 4s00_INTRO.docx (Cover page, Dedication, Document History, this Copyright notice, Table of Contents, and Introduction, including history, rules, partial solutions, etc.) 4s01_0001-0100.docx (Section I Solutions 1100) 3. Paper copies must include all pages from the two files listed in Paragraph 2 above. Paper copies must be handled in a manner similar to that described for electronic copies. For either electronic or paper copies, my solutions above 100 need not be included with the two files listed in Paragraph 2 above (solutions above 300 are still in progress as of January 2012). However, if my solutions above 100 are distributed, they must be distributed with these two files. I may, from time to time, update my various electronic files for The Four Fours Game. Please dont do it for me. If you have anything you would like to contribute, please e-mail me at [email protected] BTW, all documents are set up for two-sided printing. 4. Where I have used fonts you don't have, you may change the fonts in your own copy as needed. However, any electronic copies you distribute need to be made from my published original files. Please don't distribute paper copies from your modified files. 5. You may not sell electronic or paper copies of The Four Fours Game. You may not even ask for reimbursement for your own distribution costs. I am donating (to this point at least) my hard work and costs, and I request that you similarly donate your costs. No money should change hands in connection with distribution of The Four Fours Game, except that I reserve the right in the future to request reimbursement for my own work and costs. Fred E. Lusk III The Four Fours Game 04/24/1993~07/07/1998~01/25/2012 i 4S00_INTRO.DOCX THE FOUR FOURS GAME TABLE OF CONTENTS INTRODUCTION (1/25/2012) OBJECT 1PURPOSE 1HISTORY 2RULES 4USEFUL TABLES 10MISCELLANEOUS OPERATIONS & FUNCTIONS 11SOME PARTIAL SOLUTIONS WITH ONE FOUR 14SOME PARTIAL SOLUTIONS WITH TWO FOURS 16REFERENCES 25PARTING SHOTS 26SOLUTIONS SECTION I (1/25/2012) 1100SECTION II (1/25/2012) 101200SECTION III (1/25/2012) 201300SECTION IV Pending 301400SECTION V Pending 401500SECTION VI Pending 501600SECTION VII Pending 601700SECTION VIII Pending 701800SECTION IX Pending 801900SECTION X Pending 9011000 Fred E. Lusk III The Four Fours Game 04/24/1993~07/07/1998~01/25/2012 4S00_INTRO.DOCX [NEARLY BLANK PAGE] Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 1 4S00_INTRO.DOCX THE FOUR FOURS GAME INTRODUCTION OBJECT The object of The Four Fours Game1 is to create every whole number (i.e. positive integer) beginning with 1, using only four fours (4s) and standard mathematical operations, symbols, and notation. Simple solutions2 and clever and intricate solutions are equally worthy of note. However, Four Fours aficionados prefer the simpler solutions for aesthetic reasons. PURPOSE The raison d'tre for The Four Fours Game is that there are too many extant fours. Original research by J.A. Lusk (1984) concluded that one is indeed the loneliest number, whilst fours are found in great abundance. The purpose of this game, then, is to spread these extra fours amongst the other numbers, thus eliminating mathematical discrimination and possibly homelessness. The purpose of this treatise is fourfold: [1] to document the status of my own solutions library, [2] to provide a large number of solutions for The Four Fours Game, [3] to provide a variety of other relevant and useful information, and [4] to assist aficionados of The Four Fours Game, as well as mathematics teachers who use the game in the classroom. You know who you are. I cant compete with David Wheelers awesome The Definitive Four Fours Answer Key (see References), but I trust that my treatise will find its place in The Four Fours Game pantheon. For 1300, I provide at least three solutions;3 beyond 300, I provide at least one solution. Most of the solutions for 1100 and all of the solutions beyond 100 I developed myself.4 My solutions are not necessarily unique (see References), but I have not otherwise knowingly copied other peoples work. My solutions for each number are representative, not exhaustive, and other solutions certainly exist. Where more than one solution is provided, the simpler ones are generally shown first. The sky is the limit, but for now I am concentrating on developing solutions for 11,000, in many cases multiple solutions. 1 Or: The Four 4[']s Game, The 4 Fours Game, and The Fours Game. In the Internet age, TFFG seems appropriate. Some authors use Puzzle or Problem instead of Game, though Obsession might be a better choice. 2 That is, those using the more elementary operations, symbols, and notations. 3 For 1100, at least one solution is in the form of a square root. 4 Unfortunately, after all these years, I am unable to properly credit most of the solutions provided by others. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 2 4S00_INTRO.DOCX HISTORY The Four Fours Game has a long and lively history. In the January 1964 issue of Scientific American, Martin Gardner, author of the column Mathematical Games, provided the following history and explanation of The Four Fours Game, to which I have added a few pertinent footnotes. This particular column, which I first saw about 1978,5 was based on an imaginary conversation with Gardners friend, Dr. Matrix. It occurred to me, I said, hoping to change the subject, that because the new year ends with 4 it might be an appropriate time to introduce my readers to the old pastime of the four 4's. Do you know the game? Dr. Matrix sighed painfully. I know it well. Let me first explain the recreation. One seeks to form as many whole numbers as possible, starting with 1, by using only the digit 4 four timesno more, no lesstogether with simple mathematical symbols. Naturally one must establish what is meant by a simple symbol. This traditionally includes the arithmetic signs for addition, subtraction, multiplication, and division, together with the square root sign (repeated as many finite times as desired), parentheses, decimal points, and the factorial sign. (Factorial n is written n! It means 1 2 S n.) A decimal point may also be placed above . 4, in which case it indicates the repeating decimal . 4444 , or 49.6 The numbers 1 through 10 are easily expressed, in many different ways, by using no more than the symbols for multiplication, division, addition, and/or subtraction [see illustration]. By adding the square root sign, numbers 11 through 20 (except for 19) are readily obtained. By allowing the factorial sign and the dot used as both a decimal point and a repeating decimal sign, one can go to 112. There seems to be no way to express 1137 within these restrictions unless one employs highly bizarre combinations of the above symbols, such as the combined square root, decimal, and repeated decimal signs in the denominator of the first term in the following equation:8 4!. 4

+4. 4 The pastime was first mentioned in the issue for December 30in the palindromic, invertible year 1881of a lively London Weekly that had been founded that year by the astronomer Richard Anthony Proctor. He called his periodical Knowledge: An Illustrated Magazine of Science, Plainly WordedExactly Described. A letter to the editor expressed astonishment at the fact (shown to the writer by a friend) that all integers from 1 through 20, except 19, could be expressed by four 4's and simple signs. Factorials and dots were not allowed. Readers were asked to try their hand at it before solutions were given in a later (January 13) issue. (With the help of the factorial sign, 19 can be expressed: 4! 4 44. Can the reader of this periodical find a way to do it by using only the four arithmetical signs and the decimal point?) Since 1881 the game has enjoyed occasional revivals. A lengthy article on the topic, by W. W. Rouse Ball, appeared in the Mathematical Gazette for May, 1912, and there have been scores of subsequent articles, including tables that go above 2,000. Even now the mania will 5 Seven years after my own introduction to The Four Fours Game. 6 I use a bar over the number instead of a dot to indicate a repeating decimal. 7 Not quite true; see my solutions. 8 Combining the symbols in this way (e.g. . 4

) doesn't follow any normal mathematical convention, which is why it isnt allowed. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 3 4S00_INTRO.DOCX suddenly seize the employees of an office or laboratory, sometimes causing a work stoppage that lasts for days. Is it possible, I asked Dr. Matrix, to express 1964 with four 4's and the traditional symbols? He shook his head vigorously. Of course many important dates are possible. 1776 is 4 times 444. But 1964 is not one of them. With five 4's, yes. He jotted on my note pad: 444 +4! +4 But four 4's, no. How about 64? That, said Dr. Matrix, is not difficult. Oddly enough 64 can also be expressedunder the traditional restrictions, of coursewith three 4's and also with two. The reader is invited to try his skill on all three problems; that is, to express 64 with four 4's, with three 4's, and with two 4's. No symbols may be used other than those that have been mentioned. The task is middling hard with four 4's, ridiculously easy with three, extremely difficult with two. Next month9 I shall give the best solutions known to Dr. Matrix. Gardner also gave the following solutions with this column, except for 19, which was provided in the next issue. Note that, while each of these solutions is very simple, I did not come up with all of them myself (see Section I). Obviously, there are many ways to skin this cat. 1 = 4444 6 = 4 +4 +44 11 = 444 +4 16 = 4 +4 +4 +4 2 = 44+44 7 = 444 -4 12 = 44 +44 17 = (4 4) +44 S = 4 +4 +44 8 = 4 + 4 +4 -4 1S = 444 +4 18 = (4 4) +4 -4 4 = 4(4 -4) +4 9 = 4 +4 +44 14 = 4 +4 +4 +4 19 = 4 +4 -.4. 4 S = (4 4) +44 1u = 44 -44 1S = 444 +4 2u = (4 4) +4 +4 I first met The Four Fours Game in the fall of 1971, while in 8th grade. One morning I walked in to Algebra I to see 1 = 4444 on the chalkboard. Mr. Sherard then asked, Whats two? We didnt quite understand what he meant until someone came up with 2 = 44+44. Under Mr. Sherards guidance, we discovered the rules, then we quickly solved 3 through 10, but stalled at 11. Mr. Sherard then made it a contest: whoever solved 11 would get an ice cream at the campus canteen. This he did for each days stumper. He used the game to teach algebra and creative thinking, as well as to stimulate interest in maththus the contest. Soon, however, only two of us were still in the game: Scott Endler (3rd Period) and I (2nd Period).10 We each tried very hard to beat the other to the next solution. Sometimes Scott would win, sometimes I would. I learned about the factorial function first, so that gave me an early 9 i.e., February 1964 Scientific American. 10 Our unfair advantage was that both of our fathers were engineers and my mom had been a math major. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 4 4S00_INTRO.DOCX edge. We discovered that the prime numbers were usually the hardest, and that odd numbers were harder than even numbers. Our two most challenging solutions within the first 100 integers were 71 and 73. I remember one day waiting for the school bus and telling Scott that I had 73. He tried everything he could think of to con me out of my solution, but to no avail. I actually had two solutions, but one was based on . 4

=. 2

, which is stretching the rules since the square root operation returns a number, not a digit. Mr. Sherard eventually dropped the game as general interest waned. Personally, I found the game an ideal introduction to recreational mathematics, as well as a stimulus to my own mathematical education. Its also a great time-waster. RULES The object of The Four Fours Game is to create every positive integer beginning with 1, using only four fours (4s) and standard mathematical operations, symbols, and notation. Only fours (4) may be used as digits and numerals; no other numbers (including n and e) and no variables. Only four fours (4, 4, 4, and 4) may be used as digits and numeralsno more, no less. Only standard mathematical operations, symbols, and notation may be used, and they must be used in standard ways (e.g. the output is a number, not a digit for concatenation). The list of operations below is representative, but not exhaustive. Symbols Operations* Examples n nn nnn nnnn Numbers & Digits 4 44 444 4444 . n . nn n. n Decimal . 4 = 41u = 2S . 44 = 441uu = 112S 4.4 = 441u = 22S . n . n n. n Repeating Decimal (The overline will be used herein instead of the overdot.) . 4

= 49 4. 4

= 4u9 ( ) | ] { ] Parenthesis, Brackets, & Braces (Used for grouping. I purposely overuse these so that a detailed understanding of the order of operations is not required.) 4 |4! +(4 4)] = 224 {(4 4!) +4] 4 = 8uu Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 5 4S00_INTRO.DOCX Symbols Operations* Examples + Addition 4 +4 = 8 - Subtraction 44 -4 = 4u Multiplication 4 4 4 = 64 44 4 = 176 Also implied multiplication: (4)(4) = 16 Division (I usually use stand up fractions, but will sometimes opt for the other forms for formatting purposes.) 44 4 = 11 4444 = 111 4!. 4

= S4 x _x xy Square Root, Nested Square Roots, & Other Roots 4 = 2 _44! = 44!4= 4,u96 4 4 4.4= S2,768 4 4.4= S12 x Powers 44 = 2S6 ((4))4= 1,296 444 = 1,9S6 (44)4 = 4,294,967,296 4(44) = 1.S4u 1u154 x% x%% etc. Percent (When used as an operator, % simply means to divide by 100.) 4% = u.u4 = 12S (4!)% = u.24 = 62S 4%% = (4%)% = u.uuu4 = 12,Suu (x) Summation Function (For the special case of summing consecutive integers from 1 to x, the sigma operator can be written with just the x instead of with an index variable and upper and lower limits of summation. This is analogous to the syntax for Factorial. See also page 9.) (x) = i = x(x +1)2x=1 (4) = 4 +S +2 +1 = 1u (4 +4) = S6 ((4)) = SS (4!) = Suu (44) = 99u (444) = 98,79u (4444) = 9,876,79u Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 6 4S00_INTRO.DOCX Symbols Operations* Examples x! Factorial (See also page 9.) x! = _ix=1 = x(x -1)(x -2) S 2 1 4! = 4 S 2 1 = 24 (4 +4)! = 72u (x) Gamma Function (See also page 9.) For integers, (x) = (x -1)! (4) = S! = 6 ((4)) = 12u x Double Factorial (I first found this function in Engineering Mathematics Handbook, Jan J. Tuma, McGraw Hill. Note that the functions for odd and even numbers are not coincident. Being a Civil Engineer and not a true math expert, I dont know the origin of these functions or what they are used for. All I know is that they have nothing to do with fluid mechanics. I recall seeing somewhere a different definition for Double Factorial, but I dont remember where or what.) For x = even number: x = (2n)= 2n(2n -2)(2n -4) 6 4 2 = 2nn! = 2n(n +1) 4 = 8 |n = 2] For y = odd number: y = (2n -1) = (2n -1)(2n -S) S S 1 = 2n(n +12)n (4 . 4

) = 9 = 94S |n = S] log(x) ln(x) Common and Natural Logarithms (There are several algorithms for producing every positive integer using logarithms and nested roots, but these trivialize The Four Fours Game and are thus not permitted. Otherwise, logarithms are not very useful without the disallowed Integer, Floor, or Ceiling functions.) log(44) = 1.64S log_ 44%] = 2 ln(4) = 1.S86 olog(x) oln(x) Antilogarithms (The base 10 antilogarithms are useful.) olog(x) = 1uxolog(4) = 1u,uuu oln(x) = cxoln(4) = S4.S98 sin(x) cos(x) ton(x) etc. Trigonometric Functions (Very limited use. All trig and inverse trig functions used herein assume degrees. Radians would be useless and grads are obscure.) 44 sin __ 4. 4

%_ = 22 Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 7 4S00_INTRO.DOCX Symbols Operations* Examples orcsin(x) orccos(x) orcton(x) etc. Inverse Trigonometric Functions (A few are useful. Note that trig-1 introduces a non-4 number. For simplicity, arc will be shortened to the common a) oton _44] = 4S sinb(x), osinb(x) cosb(x), ocosb(x) tonb(x), otonb(x) Hyperbolic Functions and their Inverses (Not very useful.) cosb(. 4) = 1.u81 o bc J Determinate (Redundant) _4! 44 4_ = (4! 4) -(4 4) = 184 [nk Binomial Coefficient [nk = C(n, k) = n!(n -k)! k! C(4 ,4) = 8!(8 -4)! 4! = 7u C((4), 4) = 1S nCk C(n, k) Combination (Combinations of n objects taken k at a time.) nPk P(n, k) Permutation (Permutationsarrangementsof n objects taken k at a time.) P(n, k) = n!(n -k)! P(4 ,4) = 8!(8 -4)! = 1,68u P((4), 4) = Su B(x, y) Beta Function (Useful as a divisor. See also page 9.) B(x, y) = (x) (y)(x +y) = (x -1)! (y -1)!(x +y -1)! B[4, (4) = (2) (S)(2 +S) = 1 224 = 112 B[(4), (4) = 1168 xH0 H0(x, y) Modulus (Also known as the Remainder Function.) H0((4), (4)) = H0(Su4u,S84)= 48 H0_(4), [((4))]= H0(S84,2S1) = 1SS * The Absolute Value Function |x| is not shown because it is unnecessary. Functions such as Integer, Floor, and Ceiling are not allowed because they trivialize the game by converting non-integer approximations into integers. On the other hand, there are certainly other legal functions that are not included in this table. See USEFUL TABLES (page 10) for values for these functions Paul Bourke (see References) provides the following information regarding several related generic solutions for all integers using three and even two 4s. These can be easily converted to use four 4s, but since these solutions trivialize The Four Fours Game they are not allowed. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 8 4S00_INTRO.DOCX Any number with three 4's (contributed by Ben Rudiak-Gould) A cute solution to ANY number using just 3 fours. By admitting natural logarithms, ln( ), then any positive integer n can be represented as n = -ln jln _sqrt [sqrt((sqrt(4)) )] ln(4)[ ln(4) where the number of nested sqrt( ) functions is twice n. Or if base 4 logarithms are permitted, log4( ), then the expression becomes n = -log4_log4_sqrt [sqrt((sqrt(4)) )]_ Any number with two 4's (Contributed by whetstone) Using log10( ), any number can be represented using two fours and creative use of % as shown below: log[4 4% = log(1uu) = log(1u) 1 log[4 4%% = log(1uuuu) = log(1uu) 2 log[4 4%%% = n Some Four Fours aficionados may question the legality of certain operations listed above. As mentioned in Gardners column, only addition, subtraction, multiplication, and division were originally permitted, followed by the square root, factorial, and decimal point. I take a broader yet thoroughly consistent approach.11 So, if multiplicationwhich is really repeated additionis allowed, then certainly square roots, factorials, and decimalswhich also use simple symbols in place of longer, equivalent operationsshould be allowed as well. And, of course, they are. And, if these operations are allowed, then many other well-known and documented operations, such as powers, the Summation and Gamma Functions, percent, etc. must also be allowed. All of these valid operations are analogous to a computer: they take input and produce output. The following examples illustrate these points: Legal Operation Illegal Equivalent(s) 44+4 = 6S,SS6 44+4 4 4 4 4 4 4 4 4 44+4 4 +4 ++4 (Both equivalents use too many 4s.) . 4 and . 4

. 4 410 and . 4

49 (Both equivalents use non-4 numbers.) 4% and 4%% 4% 4100 and 4%% 410,000 (Both equivalents use non-4 numbers.) 11 Wellalmost. The Integer, Floor, Ceiling and similar functions and the log+square root tricks shown above, while technically meeting the criteria for legal operations, are excluded because they trivialize the game. We all agree that, by means of the log+square root tricks, The Four Fours Game has been definitively solved for integers from 1 to , so now lets find solutions using other methods. This is analogous to chess: just because Deep Blue defeated Garry Kasparov does not mean we shouldnt continue to play chess. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 9 4S00_INTRO.DOCX Legal Operation Illegal Equivalent(s) 4! = 24 x! = _ix=1 4! _i4=1 4! 4 S 2 1 = 24 (The first equivalent uses a variable and the second uses non-4 numbers.) (4) = 1u (x) = ix=1 (4) i4=1 (4) 4 +S +2 +1 = 1u (The first equivalent uses a variable and the second uses non-4 numbers.) (4) (x) = _ tx-1c-tJt0 (x) = (x -1)! (for integer values, x > u) (4) (4 -1)! = 6 (The definition uses varaibles and the equivalent for integers uses a non-4 number.) B(4, 4) B(x, y) = (x) (y)(x +y) = (x -1)! (y -1)!(x +y -1)! B(4, 4) (4) (2)(4 +2) = 6 112u = u.uS = 12u (The Beta Function is based on the Gamma Function, so the same comments apply.) To carry this line of thinking even further, the trigonometric and logarithmic functions are really no different, except that the symbol requires several alphabetic characters. Keep in mind that most of the operations listed above are simply abbreviations for longer calculations that would otherwise use more than the one or two 4s allotted, as well as possibly other non-4 numbers. As long as the operation is common, exists outside The Four Fours Game, and conforms to the rules and caveats stated above, there is no particular reason it should be excluded. Conversely, . 4

=. 2

(and similar) use non-standard mathematical syntax and thus are not allowed. In addition, unlike David Wheeler, I dont allow sq( ) for squaring a number because it appears to be a non-standard operator (I have only seen it used in computer programming). In Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 10 4S00_INTRO.DOCX addition, I also dont allow the binary logical operators in Wheelers impurity levels 7 and 8 because they rely on manipulating the digits of binary equivalents to base-ten integers. I am not an expert in the binary logical operators, but I suspect they can also be used to trivialize The Four Fours Game in a manner similar to the log+square root tricks shown above. USEFUL TABLES In developing solutions for The Four Fours Game, I have found it very useful to refer to tables of values for different operations and functions. Since it is possible to construct many integers with just one 4 and huge quantities of integers with two and three 4s, scanning tables of values for matches with these partial solutions can aid in the development of complete solutions.12 I have made various tables of values using Microsoft Excel and include several examples here based on these tables. You are encouraged to make your own tables because, to be truly useful, these tables need to extend well beyond what is possible to show on 8V" 11" paper. These tables arent hard to build, especially if you know how to work with the $ operator and/or array functions. Two advantages to electronic tables are automating the calculations and the ability to use the search function to find specific numbers. The first part of this section includes several tables of operations and functions, including multiplication, prime factors, powers, and the Summation, Factorial, Gamma, Double Factorial, Combination, and Permutation Functions, and the inverse of the Beta Function. Because all values for the Beta Function are less than or equal to 1, by using it as a divisor it actually works as a multiplier for the numerator. For this reason, I prefer to search for the inverse (reciprocal) of the Beta Function in concert with a multiplication table. Excel doesnt appear to have the Gamma Function, so it is necessary to use the Factorial function and the alternative definition for the Beta Function. The second and third parts of this section include tables containing numerous partial solutions containing one or two 4s. These tables are representative, not exhaustive. Partial solutions are useful for targeting regions of the whole number domain to investigate and as building blocks for complete solutions. If you need to waste a 4 for a solution, all one-4 solutions can be turned into two-4 solutions and so on. One final hint for The Four Fours Game aficionado: look for families of solutions. Consider the following related solutions: 4 +4 .4. 4 = j1921[ 4 +4 .44% = j19u21u[ 4 +4 .4. 4% = j19uu21uu[ 4 +4 4%4% = j1992u1[ 4 +4 4%. 4% = j199u2u1u[ Now, change one or both 4s to 4!, 4!!, (4), 44, or some other one- or two-4 solution and/or change . 4 to . 4

or . 4% to . 4

%, etc., and/or change the 13 to and you will discover a nearly endless bounty of additional solutions. Now, apply similar changes to solutions with different configurations, and you will have solved the entire whole number domain without any tricks. 12 For example, the square root solutions for the larger integers in the 1100 range. 13 I used instead of _ so that the operations visually correspond to the results, which have the smallest on top. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 11 4S00_INTRO.DOCX MISCELLANEOUS OPERATIONS AND FUNCTIONS Multiplication and Prime Factors The following table provides values for multiplication up to 12 24 and prime factors up to 24. I suggest making a multiplication table up to at least Su Su (my Excel table is Suu Suu and could be made even larger) and a table of prime factors up to at least 300. Piime Factois n 1 2 S 4 S 6 7 8 9 1u 11 12 1 1 2 S 4 S 6 7 8 9 1u 11 122 2 2 4 6 8 1u 12 14 16 18 2u 22 24S S S 6 9 12 1S 18 21 24 27 Su SS S622 4 4 8 12 16 2u 24 28 S2 S6 4u 44 48S S S 1u 1S 2u 2S Su SS 4u 4S Su SS 6u2 S 6 6 12 18 24 Su S6 42 48 S4 6u 66 727 7 7 14 21 28 SS 42 49 S6 6S 7u 77 8423 8 8 16 24 S2 4u 48 S6 64 72 8u 88 96S2 9 9 18 27 S6 4S S4 6S 72 81 9u 99 1u82 S 1u 1u 2u Su 4u Su 6u 7u 8u 9u 1uu 11u 12u11 11 11 22 SS 44 SS 66 77 88 99 11u 121 1S222 S 12 12 24 S6 48 6u 72 84 96 1u8 12u 1S2 1441S 1S 1S 26 S9 S2 6S 78 91 1u4 117 1Su 14S 1S62 7 14 14 28 42 S6 7u 84 98 112 126 14u 1S4 168S S 1S 1S Su 4S 6u 7S 9u 1uS 12u 1SS 1Su 16S 18u24 16 16 S2 48 64 8u 96 112 128 144 16u 176 19217 17 17 S4 S1 68 8S 1u2 119 1S6 1SS 17u 187 2u42 S2 18 18 S6 S4 72 9u 1u8 126 144 162 18u 198 21619 19 19 S8 S7 76 9S 114 1SS 1S2 171 19u 2u9 22822 S 2u 2u 4u 6u 8u 1uu 12u 14u 16u 18u 2uu 22u 24uS 7 21 21 42 6S 84 1uS 126 147 168 189 21u 2S1 2S22 11 22 22 44 66 88 11u 1S2 1S4 176 198 22u 242 2642S 2S 2S 46 69 92 11S 1S8 161 184 2u7 2Su 2SS 27623 S 24 24 48 72 96 12u 144 168 192 216 24u 264 288 Powers and the Summation, Factorial, Gamma, and Double Factorial Functions The following table provides values for integer powers up to 4 and the Summation, Factorial, Gamma, and Double Factorial Functions, all for n from 1 to 24. The largest values are presented in calculator-style scientific notation. Summations and squares for n > 24 also prove to be especially useful (my Excel table goes to 300). The larger values for the other functions are generally not useful for generating solutions for small integers, though there are exceptions.14 14 For example, the two-4 solution for 64 based on 424. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 12 4S00_INTRO.DOCX n n2 n3 n4 L(n) n! I(n) n 1 1 1 1 1 1 1 12 4 8 16 S 2 1 2S 9 27 81 6 6 2 S4 16 64 2S6 1u 24 6 8S 2S 12S 62S 1S 12u 24 1S6 S6 216 1,296 21 72u 12u 487 49 S4S 2,4u1 28 S,u4u 72u 1uS8 64 S12 4,u96 S6 4u,S2u S,u4u S849 81 729 6,S61 4S S62,88u 4u,S2u 94S1u 1uu 1,uuu 1u,uuu SS S,628,8uu S62,88u S,84u11 121 1,SS1 14,641 66 S9,916,8uu S,628,8uu 1u,S9S12 144 1,728 2u,7S6 78 479,uu1,6uu S9,916,8uu 46,u8u1S 169 2,197 28,S61 91 6.227Eu9 479,uu1,6uu 1SS,1SS14 196 2,744 S8,416 1uS 8.718E1u 6.227Eu9 64S,12u1S 22S S,S7S Su,62S 12u 1.Su8E12 8.718E1u 2,u27,u2S16 2S6 4,u96 6S,SS6 1S6 2.u92E1S 1.Su8E12 1u,S21,92u17 289 4,91S 8S,S21 1SS S.SS7E14 2.u92E1S S4,4S9,42S18 S24 S,8S2 1u4,976 171 6.4u2E1S S.SS7E14 18S,794,S6u19 S61 6,8S9 1Su,S21 19u 1.216E17 6.4u2E1S 6S4,729,u7S2u 4uu 8,uuu 16u,uuu 21u 2.4SSE18 1.216E17 S.716Eu921 441 9,261 194,481 2S1 S.1u9E19 2.4SSE18 1.S7SE1u22 484 1u,648 2S4,2S6 2SS 1.124E21 S.1u9E19 8.17SE1u2S S29 12,167 279,841 276 2.S8SE22 1.124E21 S.162E1124 S76 1S,824 SS1,776 Suu 6.2u4E2S 2.S8SE22 1.6SSE12 Combination Function The following table provides the number of combinations up to n = 12 and k = 8. I suggest making a table for at least n = 2u. My Excel table goes to n = S6 by k = S6. The Combination Function is symmetrical about k = n 2 . n 1 2 S 4 S 6 7 8 9 1u 11 12 k 1 1 2 S 4 S 6 7 8 9 1u 11 12 2 1 S 6 1u 1S 21 28 S6 4S SS 66 S 1 4 1u 2u SS S6 84 12u 16S 22u 4 1 S 1S SS 7u 126 21u SSu 49S S 1 6 21 S6 126 2S2 462 792 6 1 7 28 84 21u 462 924 7 1 8 S6 12u SSu 792 8 1 9 4S 16S 49S Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 13 4S00_INTRO.DOCX Permutation Function The following table provides the number of permutations up to n = 12 and k = 8, but only for values less than 100,000 (to save space, larger values are denoted with +++). I suggest making a table for at least n = 2u. My Excel table goes to n = S6 by k = 12. However, the results for k 8 are so large that I havent found a use for themyet. Unlike the Combination Function, the Permutation Function is not symmetrical. n 1 2 S 4 S 6 7 8 9 1u 11 12 k 1 1 2 S 4 S 6 7 8 9 1u 11 122 2 6 12 2u Su 42 S6 72 9u 11u 1S2S 6 24 6u 12u 21u SS6 Su4 72u 99u 1,S2u4 24 12u S6u 84u 1,68u S,u24 S,u4u 7,92u 11,88uS 12u 72u 2,S2u 6,72u 1S,12u Su,24u SS,44u 9S,u4u6 72u S,u4u 2u,16u 6u,48u +++ +++ +++7 S,u4u 4u,S2u +++ +++ +++ +++8 4u,S2u +++ +++ +++ +++ Inverse Beta Function The following table provides values for the inverse (reciprocal) of the Beta Function up to x = 12 and y = 8, but only for values less than 100,000 (to save space, larger values are denoted with +++). I suggest making a table for at least x = 2u. My Excel table goes to x = S6 by y = 12. However, the results for k 8 are so large that I havent found a use for themyet. The Beta Function and its inverse are symmetrical about y = x 2 . x 1 2 S 4 S 6 7 8 9 1u 11 12 y 1 1 2 S 4 S 6 7 8 9 1u 11 122 2 6 12 2u Su 42 S6 72 9u 11u 1S2 1S6S S 12 Su 6u 1uS 168 2S2 S6u 49S 66u 8S8 1,u924 4 2u 6u 14u 28u Su4 84u 1,S2u 1,98u 2,86u 4,uu4 S,46uS S Su 1uS 28u 6Su 1,26u 2,S1u S,96u 6,4SS 1u,u1u 1S,u1S 21,84u6 6 42 168 Su4 1,26u 2,772 S,S44 1u,296 18,u18 Su,uSu 48,u48 74,2S67 7 S6 2S2 84u 2,S1u S,S44 12,u12 24,u24 4S,u4S 8u,u8u +++ +++8 8 72 S6u 1,S2u S,96u 1u,296 24,u24 S1,48u +++ +++ +++ +++ I also suggest making (or finding) a table of useful trig and inverse trig values. For example, oton(1) = osin(1 2 ) = ocos(1 2 ) = 4S and osin(u.S) = ocos [S 4 = Su, etc. Such tables can be found in Schaum's Outline of Mathematical Handbook of Formulas and Tables and Engineering Mathematics Handbook, by Jan J. Tuma. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 14 4S00_INTRO.DOCX SOME PARTIAL SOLUTIONS WITH ONE FOUR Fraction Decimal Equation 125,000 u.uuuu4 . 4%%122,500 u.uuuu444

. 4

%%12500 u.uuu4 4%%1250 u.uu4 . 4%1225 u.uu444

. 4

%1150 u.uu666

. 4

%% = [. 4

%1100 u.u1 [(4) %150 u.u2 (4)%3100 u.uS [(4) %125 u.u4 4%350 u.u6 ((4))%115 u.u666

. 4

%15 u.2 4%625 u.24 (4!)%25 u.4 . 449 u.444

. 4

23 u.666

. 4

My partial one-4 solutions (fractions) Integer Equation 1 (4)2 4S (4)4 4 6 (4) = [(4)8 41u (4) = olog[(4)21 ((4))24 4!Su ocsc(4)S6 (4)4S oton [(4)48 ((4))SS ((4))6u oscc(4)9u osin [(4)1uu olog(4)12u ((4))1SS15oton [-(4)2S1 [((4))Suu (4!)S84 (4)46S [ocsc(4)666 ((4)) 15 The trig functions are cyclical; 135 is the first positive value for oton(-1). I use the first positive value for all trig functions. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 15 4S00_INTRO.DOCX Integer Equation 72u ((4))! 1,uuu olog[(4)1,uSS _oton [(4)]1,176 [((4))1,S4u [((4))1,8Su [oscc(4)S,84u ((4)) 4,u9S _osin [(4)]S,u4u (4) S,uSu [olog(4)7,26u [((4))9,18u _oton [-(4)]1u,uuu olog(4) 26,796 _[((4))]4u,S2u (4)! 4S,1Su ((4!)) 7S,92u ((4)) My partial one-4 solutions (integers) Integer Equation NOTES Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 16 4S00_INTRO.DOCX SOME PARTIAL SOLUTIONS WITH TWO FOURS The large number of one-4 solutions in the preceding tables (which, by the way, are not exhaustive) leads to an impossibly large number of two-4 solutions. Consequently, the following tables of two-4 solutions just scratch the surface. However, I was able to make nearly all of the integers from 1 to 100 with just two 4s. I also paid special attention to creating odd numbers, which are often harder to come by than even numbers. With two exceptions, this table ends at 10,000; it will be extended in the future. Suggestions for making additional two-4 solutions by combining two one-4 solutions from the preceding tables include: Combine integers and fractions in various ways. I have provided several examples using 4 as the integer. The results should be useful for multiplying and dividing with other numbers to make complete solutions. Combine two fractions in various ways. I have provided more than a dozen examples. These should be useful as divisors for making complete solutions. Fraction Decimal Equation 12,500 u.uuu4 [(4)%411,000 u.uu1 [(4) %(4)1625 u.uu16 (4%)(4%)1500 u.uu2 (4)%(4)1375 u.uu2666

. 4

% 4%112,500 u.uu44 . 44%1168 u.uuS9S2S B[(4), (4)1100 u.u1 _44] %4250 u.u16 (. 4)(4%)150 u.u2 4%4275 u.u2666

. 4

%-4%17300 u.uS666

. 4

%-[(4) %23300 u.u7666

. 4

%+[(4) %112 u.u8SSS

B[4, (4)875 u.1u666

. 4

%+4%425 u.16 (. 4)(. 4)1690 = 845 u.1777

(. 4)(. 4

)Fraction Decimal Equation 1681 u.197SS (. 4

)(. 4

)59250 u.2S6 (4!)%-.4%61250 u.244 (4!)%+.4%830 = 415 u.2666

. 4 . 4

827 u.296296

. 4

. 4

391900 u.4S444

. 4

-[(4) %1125 u.44 . 44409900 u.4S444

. 4

+[(4) %12 u.S sin [4 . 4

%154225 u.68444

(4!)%+. 4

3845 u.8444

. 4+. 4

89 u.888

. 4

+. 4

Examples of fractions > 1 (based on 4) 85 1.6 4 .4169 1.777

4 . 4

83 2.666

4 . 4

103 S.SSS

4 -. 4

329 S.SSS

4-. 4

185 S.6 4 -.4 Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 17 4S00_INTRO.DOCX Fraction Decimal Equation 9925 S.96 4 -4%10125 4.u4 4 +4%225 4.4 4.4409 4.444

4. 4

143 4.666

4 +. 4

503 16.666

4(4!)%2003 66.666

4((4))%4003 1SS.SSS

4[(4) %My partial two-4 solutions (fractions) Fraction Decimal Equation NOTES Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 18 4S00_INTRO.DOCX Integer Equation 1 44 2 4 -4 = 44S _4. 4

= (4)44 4 +4 = 4 4S 4. 4 6 4 +4 = 4!47 4 -(4) 8 4 +4 9 4. 4

= 4 +(4)1u 4. 4 = (4) +411 4 +(4)12 4!4 = 4 (4) = 4 +41S (4) +(4)14 (4) +4 1S _4. 4_ = _4. 4_16 4 4 = 4417 ((4)) -418 4. 4

= (4) +419 ((4)) -42u 4! -4 = 4. 421 4! -(4) = (4 +4)22 4! -4 Integer Equation 2S ((4)) +424 (4 +4)! = 4 (4)2S olog(4)4 = ((4)) +426 4! +427 4! +(4) = ((4)) +(4)= [(4)Z(4)28 4! +4 = C(4, 4)= [4 -(4)29 ((4)) +4= ocsc(4) -(4)Su osin _44 _ = _ 4. 4

%S1 ((4)) +(4)= ocsc(4) +(4)S2 4 4 = 4.4= 4! +4SS ocsc(4) +(4)S4 4! +(4) = ocsc(4) +4SS (4) -(4)S6 4!. 4

= (4 +4) = ((4))4S7 (4) +(4)S8 (4) +4S9 (4) +(4)4u (4) +4 = _(4)(4!)%= ((4)) -4 = .4% olog(4)= 4 olog(4) Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 19 4S00_INTRO.DOCX Integer Equation 41 oton [(4) -442 (4) +(4)= oton [(4) -(4)4S oton [(4) -444 44 4S oton _44] = _4. 4

]46 oton [(4) +(4)47 oton [(4) +448 4! +4! = 4 (4)49 oton [(4) +4Su 44% = olog(4)4S1 ((4)) -4S2 ((4)) +4= oscc(4) -4SS ((4)) -4S4 4!. 4

SS _4. 4] = [(4 4)S6 ((4)) +4 = (4)!((4))!= C [4, (4)S7 ((4)) +4S8 ((4)) +(4)= oscc(4) -4S9 ((4)) +46u 4!. 4 Integer Equation 61 oscc(4) +(4)62 oscc(4) +46S oscc(4) +(4)64 __44! = 4Z(4) = (4)(4)6S ((4)) +(4)66 [4 +(4)67 68 oscc(4) +469 oton [(4) + 4!7u C(4 ,4)71 72 ((4)) +4! = (4) 47S 74 7S (4) 4%76 ((4)) +((4))77 78 _4!4]79 ((4)) + 4!8u (4) 481 [(4)482 osin [(4) -48S 84 ((4)) 48S ((4)) +ocsc(4)86 osin [(4) -4 Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 20 4S00_INTRO.DOCX Integer Equation 87 osin [(4) -(4)88 osin [(4) -489 osin [(4) -(4)9u . 4. 4

% = osin _44]91 [(4) +(4)= osin [(4) +(4)92 osin [(4) +49S osin [(4) +(4)94 osin [(4) +49S 96 4! 4 97 olog(4) -(4)98 olog(4) -499 olog(4) -(4)1uu 44% = olog_ 44] = ((4))41uS (4)!(4) = [4 -(4) = ((4) +4)111 [((4)) -((4))112 (4)oton [(4)12u _4. 4_! = __4. 4__1S1 [((4)) -olog(4)1SS oton _-44]1S6 (4 4) Integer Equation 141 [((4)) -osin [(4)144 4! (4) = (4) 4 = ((4)) (4) 1Su . 4

. 4

%1SS (4) - [((4))1S4 . 4

[((4)) 16S ((4)) (4)168 ((4)) 4171 ((4) +4)18u 4 oton [(4)18S [((4)) -((4))19S [((4)) -(4)2uu 44% = 4 olog(4)2u1 [((4)) -ocsc(4)2u7 [((4)) -4!21u (4! -4) = (4)4!= C [olog [(4) , 4216 ((4))Z(4)22u 4 ((4))2S1 [(4 +4)24u 4! (4)249 (4) -oton [-(4)2S6 4427u ((4)). 4

Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 21 4S00_INTRO.DOCX Integer Equation 276 C(4!, 4) 296 . 4

((4)) S29 (4) -((4))Suu ((4)). 4 SSu ((4)) (4)SS6 (4)!((4)) = P [4, (4)SS9 (4) -oton [(4)S4S [ocsc(4) -((4))S6u oscc(4) (4) = P((4), 4) S84 (4 +4) 4SS C(ocsc(4), 4) 441 [((4))4444 . 4

((4)) 4Su 4. 4

% 462 4 [((4))Suu 4. 4% Su4 4! ((4))S12 (4)Z(4) SS2 P(4!, 4) S76 (4!)4 6uu 4!4% 6Su ocsc(4) ((4))67S (4). 4

% Integer Equation 69S (4) [((4))72u _4!4] !729 [(4)I(4)7Su (4). 4%7S6 (4) ((4))87u P(ocsc(4), 4) 896 (4)!oton [(4)9uu 4. 4

%924 4 [((4))94S _4. 4

] 99u (44)999 ((4)). 4

1,uuu 4. 4%1,uu8 ((4)) ((4))1,u24 44%= (4)Z(4)1,u8u ((4))!. 4

1,1SS ((4)) ((4))1,26u oscc(4) ((4))1,296 ((4))41,SSu C [((4)), (4) 1,SSu (4). 4

% Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 22 4S00_INTRO.DOCX Integer Equation 1,Suu (4). 4% 1,62u ((4))!. 4

1,68u P(4 ,4) 1,8uu 4. 4

% 2,uuu 4. 4% 2,u24 C [4!, (4) 2,u2S _oton [(4)]42,2Su (4). 4

% 2,Su4 [((4))42,Suu (4). 4% 2,64u ((4)) ((4)) 2,88u 4 ((4))!S,u2S [((4))4S,6uu [oscc(4)4S,84u _4. 4] 4,u96 (4)4 = 4I(4) = _44!S,4uu 4!. 4

% S,98S C(((4)), 4) 6,uuu 4!. 4% 6,S61 [(4)4 7,776 (4).4 Integer Equation 7,98u P [((4)), (4) 8,1uu _osin [(4)]4 9,uuu . 4. 4

%%9,261 [((4))Z(4)1u,uuu olog(4) = ((4))41u,S9S [4 +(4) 46,u8u (4 +4)My partial two-4 solutions (integers) Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 23 4S00_INTRO.DOCX Integer Equation Integer Equation Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 24 4S00_INTRO.DOCX Integer Equation Integer Equation NOTES Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 25 4S00_INTRO.DOCX REFERENCES There are many websites devoted to The Four Fours Game. The following list is representative, not exhaustive. The sites with the most solutions are bolded. Bogomolny, Alexander Representation of numbers with four 4's http://www.cut-the-knot.org/arithmetic/funny/4_4.shtml Bourke, Paul Four Fours Problem http://paulbourke.net/fun/4444/ Carver, Ruth Four 4's Puzzle http://mathforum.org/ruth/four4s.puzzle.html Forman, Ben The Four 4's Project https://docs.google.com/viewer?a=v&q=cache:21s6joFz2loJ:s89940423.onlinehome.us/images/6/61/The_Four_4%25E2%2580%2599s_Project.doc+four+4's+solutions&hl=en&gl=us&pid=bl&srcid=ADGEESj8O-EA0wxXsn3puBqewGqwkJHGv0KPPeQdrq1pV3MPZEN2myWrma3Pcr7xP8COFEHAuXkVFfeIP0nRBKsPLPiliNu5P39WzVV4FzQo5T81y2zVQNRg7R6LrHfPbMDF1ZzEms8P&sig=AHIEtbTRpcGxfJaBDi0GPwTk8uknFQad4A Hammond, Nicholas Four Fours Problem http://home.comcast.net/~nicolas.hammond/fun/four_fours.html Johansen, Carl The Four Fours http://www.carljohansen.co.uk/fourfours/ Marxen, Heiner The Year Puzzle http://www.drb.insel.de/~heiner/Puzzles/Year/R4444 MersenneForum.org Classic Problem: Four 4's http://www.mersenneforum.org/showthread.php?t=4756 SciMath Newsgroup Help With Four 4's http://www.math.niu.edu/~rusin/uses-math/games/krypto/4fours Wheeler, David A. The Definitive Four Fours Answer Key http://www.dwheeler.com/fourfours/ Wheels.org The Four Fours Problem http://wheels.org/math/44s.html Wikipedia.org Four Fours http://en.wikipedia.org/wiki/Four_fours Valentine, John About Four Fours http://johnvalentine.co.uk/fourfours.php Wilson, Steven J. Intergermania http://www.milefoot.com/math/integermania/four4s-1.htm Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 26 4S00_INTRO.DOCX PARTING SHOTS Here are some extra 4's for you to use (4 +4 +4)4 to be exact and in 4! pt size. Of course, if you dont have the same font catalog, your results will vary. 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 G 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 44 44 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 44 4 4 4 4 4 4 44 4 4 4 4 4 44 4 4 4 4 444 4 44 4 4 4 4 4 4 44 4 4May the fours be with you. Beware the dark side of the fours. Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 27 4S00_INTRO.DOCX Fred E. Lusk III The Four Fours Game Introduction 04/24/1993~07/07/1998~01/25/2012 28 4S00_INTRO.DOCX [NEARLY BLANK PAGE] Fred E. Lusk III The Fours Game 1 to 100 04/24/1993~06/30/1998~1/25/2012 Page I.1 4s01_0001-0100.docx SECTION I SOLUTIONS 1100 Hang on, here goes1 1 4444 = _4444 4. 4- 4. 4

4 +4[4!44444 4 - 4 - 4 - (4) 4! + 4 - ((4)) -(4)4 - 4(4) - 4 ((4)) - ((4))4! + (4) 4 olog(4) 4%42 44+ 44 4 + 4 - 4 4_44,4_4!4 + 4 + 4 4. 4 - _4. 4

_4. 4- 4!44! + 4! - 44 __4(4+4)4 4! + 4(4) + 4S 4 + 44- 4 ___4. 4

] _4. 4

] 4!4 + 4 + 44! 44.4 ((4))!4 _4. 4_! ((4) + 4)(4) - (4)4 4 + 4 - 4 - 4 4 + 4 + 4 + 4 4!4 + 4 + 4(4 + 4 - 4)4 4! + 4! -44 44(4) + (4). 4 ((4) + 4)4 _4! + (4) + 4 - 4 44 - _(4)(4!)%__((4))[Z(4)+ (4) + 4 ___(4 4)(4+4) 1 The number of solutions provided for a particular integer most often reflects the amount of effort expended (or even dumb luck at stumbling across a solution) rather than the degree of difficulty. The solutions for 31, 33, 34, 36, and 38 that are flagged with *** are those solutions published in the Winter 1984 edition of The Bent (the quarterly magazine of Tau Beta Pi, the engineering honor society) that I had not independently developed. Fred E. Lusk III The Fours Game 1 to 100 04/24/1993~06/30/1998~1/25/2012 Page I.2 4s01_0001-0100.docx SECTION I SOLUTIONS 1100 4 (cont) _4Z(4)44 4!4- 4 - 4 (4)(4) - 4!. 44. 4 -(4 4) 4!. 4 _4. 4_ 44%- (4! 4)(4)!olog(4) + ((4) 4) 44 4Z(4) ((4))! -((4)) - ((4))((4)) [olog(4) - (4) - 4 - 4S 4 + 4 + 44 4!4 - 44 4!4% _4. 4_!_(4 + 4)! + 44 (4 4) +44 __4. 4_ + 4. 44. 4% 44% 6 4 +4 +4 - 4 4 + 4 + 44 4!. 4- 4!. 4

4! 44 + 4 4! +4 +4 +4 ocsc(4)4 + 444!4% 44% P((4), 4) 4!. 47 4 + 4 - 44 4!4 + 44 4. 4

- 4 + 4_4! + 4! + 44 ((4))4 - 44 C(4 ,4) . 448 4 + 4 + 4 - 4 4 + 4 + 4 + 4 (4 4) - 4 - 44!4 - 44 4! 44 +4 _(4 4) + 4.4((4))! + ((4) 4)olog(4) ((4))(4 4) - (4) 44%- 44!. 4 (4) 44 Fred E. Lusk III The Fours Game 1 to 100 04/24/1993~06/30/1998~1/25/2012 Page I.3 4s01_0001-0100.docx SECTION I SOLUTIONS 1100 9 4 + 4 + 44 4!(. 4

)(4 + 4)4!4_44!_((4) 4) + 44 4. 4

44 _4 -44]41u 4 +4 +4 - 4 4 + 4 +4 + 4 4! - 4. 4 -44!(. 4)(4 + 4) 44% 4. 4 _[(4! + 4) 4 - 4_4. 4_! B[4, (4)11 4. 4+ 44 4. 4 + 4 + 4 4!4 - 44__4. 4_! + 44 4 + 4 + _4. 4

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_ -4 Fred E. Lusk III The Fours Game 1 to 100 04/24/1993~06/30/1998~1/25/2012 Page I.21 4s01_0001-0100.docx Solutions History 1100 # 199S 1998 2u12 1 S S 1u 2 S S 9 S S S 6 4 S S 21 S S S 7 6 S S 8 7 S S 6 8 S S 1u 9 S S 6 1u S S 7 11 S S 6 12 S S 8 1S 6 6 9 14 S S 6 1S S S 7 16 S S 1S 17 6 6 7 18 S S 6 19 S S 8 2u S S 7 21 4 4 8 22 S S 8 2S S S 8 24 4 4 17 2S S S 1u 26 S S 7 27 S S S 28 S S 7 29 S S 4 Su S S 6 S1 S S S S2 S S 11 SS 4 4 7 S4 S S 7 SS 4 4 7 S6 S S 17 S7 4 4 6 S8 S S S S9 S S S 4u S S 7 # 199S 1998 2u1241 S S S42 S S 64S S S S44 S S 84S S S 746 S S 747 S S S48 S S 849 S S 7Su S S 8S1 S S SS2 S S 4SS S S SS4 S S 4SS S S 7S6 S S SS7 S S 4S8 4 4 7S9 4 4 76u S S 961 S S S62 S S 46S S S 864 4 4 1S6S S S 766 S S 767 S S 768 S S 769 S S S7u 4 4 871 6 6 972 6 6 1u7S 4 4 874 S S S7S S S 876 S S S77 S S S78 S S S79 S S 48u 4 4 9# 199S 1998 2u1281 4 4 982 S S S8S S S S84 4 4 78S S S S86 S S 487 4 4 888 S S 689 4 4 79u 4 4 691 S S 892 S S 69S S S S94 4 4 S9S S S 796 S 4 2S97 S S 798 4 4 899 4 4 81uu S S 14L S4S SS4 741 Paitial Solutions*#4s 199S 1998 2u120ne 28 28 S8Two 77 77 287L 1uS 1uS S4S* Boes not incluue tables of Niscellaneous 0peiations anu Functions, which aie new to the 2u12 publication. Fred E. Lusk III The Fours Game 1 to 100 04/24/1993~06/30/1998~1/25/2012 Page I.22 4s01_0001-0100.docx |NEARLY BLANK PAuE] Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.1 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 1u1 44%+ 44 (4! 4) + 4. 4 _olog(4) + 44% +(4)(4) +4 +.4. 4 1u2 44% + 44 (4! 4) + 4 + 4 (4)!(4) - _4. 4

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44 - .4. 4 4! + 4!+. 4

. 4

oton _44] + (4)(4) _P _4!4, 4] + (4)11u 44%+ 4. 4 (4)!(4) + 4. 4 [4 + (4) 4. 4_4! 4. 4_ - (4) 44 _4. 4] _4. 4_! - 4. 4__((4)) +44] olog(4) _4 _ [olog(4) + 4. 4%]_4 _ _4. 4]_4 _4 __oton [(4)]4+ olog[(4)_111 4444 44.4. 4 (4)!(4)+ 4 + 444%+ 4 + (4) (4)oton [(4)- 44112 (4! 4) + (4 4) 44. 4 + 4 _4. 4_! - 4 - 444% + 4!4 11S (4)!(4)+ 4 + 4 ((4)) - 4 - _4. 4

(4)oton [(4)+ 44_4 (4)!((4))!_ + (4) __44! + oton [(4) + 4 Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.3 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 114 44. 4 + 4 _4. 4_! - 4 - 4 C(4 ,4) + 4411S _4. 4_! - 4. 4 (4)!(4)+ 4. 4 . 4. 4

%+ olog(4)4__oton [-(4)] + _osin [(4)] - 44%116 44%+ (4 4) (4! 4) + 4! - 4 44 - 4!4__ [((4))]4 + ((4)) + (4)117 _4. 4_! - _4. 4

4! + 4. 4

4 (4)!(4) + 4!4118 (4! 4) + 4! -4 _4. 4_! - 44 (4! (4)) - 4444%+ 4. 4

((4)) + 4+. 4

. 4

119 _4. 4_! - 44 4! + 4! - .4. 4 (4)!(4)+ (4) + 412u 4 (4! + 4 + 4) 4!4 4. 4 _ 44% 4! (4)44%+ 4. 4 _olog(4) + 4!. 4

%- olog [(4)121 _4. 4_! + 44 (4! -4)44 4! + 4! + .4. 4_(4) + 44]4 44%+ (4 + 4) (4)!(4)+ (4 4)____4. 4_4!- 4 Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.4 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 122 (4! 4) + 4! +4 _4. 4_! + 44 44%+ 4! - 4____4. 4_4!- (4) 12S _4. 4_! + _4. 4

44. 4

+ 4! (4)!(4)+ 4. 4

____4. 4_4!-4 124 (4! 4) + 4! + 4 444 - 4 ((4) 4) + 444. 4%- 44 __44! + 4!. 4 4 _(4)(4) -4__[(4) (4) +4412S _4. 4_! + 4. 4 ___[4 +444!__4. 4_(4+4)C(4 ,4) + _4. 4] 126 44 - 44 44%+ 4! + 4 (4 4Z(4)) - 4127 44 - 44 4. 4%+ 44 (4 4) - 4. 4

((4)) +4+. 4

. 4

128 (4! + 4 + 4) 4 44 44 ___(4)(4!+4!+4) Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.5 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 129 44 + 44 _4. 4_! + 4. 4

[44 - (4) (4)1Su 44 + 44 _4. 4_! + 4. 4 (4)!(4)+ olog(4)41S1 (4). 4% 4 + (4) (4)!(4) +4! + 4 [(4 + 4) - 44%((4)) + 4 + .4. 4 1S2 444 + 4 44%+ 4! + 4 4 _4! + 4. 4

]1SS (4). 4% 4 + 4 [4 olog(4) - (4)(4) (4)!(4)+ 4! + 41S4 44. 4 + 4! . 4

. 4

%- (4 4) C(4!, 4) - 441SS 4. 4 + .44% 4!4%- oscc(4)4 44 oton _-44](4)!(4) +osin _44 _1S6 _4. 4_! + (4 4) 4 _4! + 4. 4] 44%+ 4!. 4

_4!4]4- 4 4 _(4)(4) + 4_ 4!. 4+. 4

. 4

1S7 4. 4%+ ((4))4 C(4!, 4) - 44 (4)!(4) +(4 4)_oton [(4) _4. 4

_ + 4 osin [(4) + ((4)) - 441S8 . 4

. 4

%- 4!4 44%+ (4) + 4 ((4)) + (4 4) + 4[4 ((4)) +4! + 4 (44 + 4) (4) Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.6 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 1S9 (4! (4)) - 4. 4 ((4)) +4! - 4. 4 (4)!(4)+ 4! + (4)44%+ oton [(4) - (4)14u _4. 4_! + 4! - 4 _4!4]4- 4 _44!. 4

- 4olog(4) + 44 - 4 ((4)) + ((4)) + 4Z(4) 141 ((4)) + 4! - _4. 4

(4! 4) + oton _44] osin [44 + 4. 4

[oton [(4) + 4 _4. 4

((4)) -olog(4) -44(4)!((4))!+ .4. 4 44%+ oton [(4) -4142 _4. 4_! + 4! - 4 _4!4]4- 4 _44!. 4

- 4((4)) - olog(4) + 44 (4) - 44%4 14S _44!-. 4

. 4

((4)) + 4! - 44 olog(4) + 44 - (4)_4!4]4- (4) ((4)) - olog(4) + (4)4 144 (4 + 4 + 4)4 44%+ 44 4!4 4!44Z(4). 4

. 4

. 4 (4 + 4)!4 . 44. 4

%+oton [(4)(4 + 4) (4 + 4) [((4)) - 4!4 + 4 (((4)) 4) + 4!. 4_(4)! + [((4)) - 4!4 _(4)!4 +(4!)4 Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.7 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 14S 44%+ _4. 4

] _44!+. 4

. 4

((4)) + 4! + 44olog(4) + 44 + (4) [ocsc(4) - (4!) - 4! + 4146 _4. 4_! + 4! + 4 _4!4]4+ 4 _44!. 4

+ 4((4)) + . 4

. 4

4 olog(4) + 44 + 4147 . 4

. 4

%- _4. 4

((4)) + 4! + _4. 4

_((4)) + 44_ (4)(4)!(4) + oton [(4) - (4) [(4 + 4) - (((4)) 4)148 _4. 4_! + 4! + 4 _4!4]4+ 4 _44!. 4

+ 444% +4! + 4! 4Z(4) + ((4)) + (4) (4 4) + 4!4 149 . 4

. 4

%- 44 ((4)) +4! + 4. 44!4%- 44(4)!(4)+ 44 [oscc(4)4-4!4!1Su 4!4% (4 +4) 44%. 4

. 4

4! 4(4Z(4))%4 4. 4

(4!)% _4. 4_ 4. 4 _4! + 44] (4)4 + 4 + 44% osin _44] +oscc _ 44] ((4)) + 4! + 4 + 444%+ ((4)) + 4 _44. 4

%% Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.8 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 1S1 . 4

. 4

%+ 44 4!4 4%+ (4) [oscc(4)4+4!4!4!. 4+ .4. 4 44%+ ((4)) - 4((4)) + 4! + 4 - (4)1S2 (44 4) - 4! 4!. 4 .4 +4 [oscc(4)4+ ((4))4!1SS . 4

. 4

%+ _4. 4

((4)) + 4! + 4. 4

44%+((4)) -41S4 4!. 4 .4 + 4 (4! (4)) + 4. 4 (4!) + 4 + 44oscc(4) + 4+. 4

. 4

1SS . 4

. 4

%+ 4. 4 4Z(4) - 4. 4 (4!) +(4) + 4444%+ _4. 4] oton _-44] +4. 41S6 (4 + 4)!4 - 4! (4! (4)) + 4!4 (4!) + 4!441S7 _4 oton [(4)] - 4! + (4) [oscc(4) ((4)) - 44[((4)) (4) - 4 (4)!(4)+ oscc(4) -41S8 . 4

. 4

%+ 4 + 4 (((4)) 4) - 4. 4 (4!) + (4 4)41S9 . 4

. 4

%+ 4. 4

osin [(4) + ((4)) + oton [(4) - 4!_44! - .4. 4 (4!) +4!!. 4

4 Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.9 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 16u 4 4 4. 4 . 4

. 4

%+ 4. 4 . 4

(4 + 4)!4_ _ [((4))] - [((4)) - 4. 4161 _44! + .4. 4 ((4))4- 44 ((4)) + 4!-. 4

. 4

162 4!. 4

_4. 4

((4))44 + 4 (4 + 4) + 4!. 4

16S ((4)) + 4!+. 4

. 4

((4))4+ 44 (4!) +4! + 44164 _4. 4_! +44 44%+ 4Z(4) ((4 +4)) - (4)4_ _ [(4 + 4)] + 44%16S 4Z(4) + 4. 4 ((4)) (4 + 4)4 (4!) + osin _44 _4166 . 4

. 4

%+ (4 4) 44 - . 4. 4

%(4)!((4))- 44((4)) + 44 +4 . 4 _[ocsc(4) - 44%_[(((4)) 4) -(4) 4 olog(4) +4Z(4) + 4167 _4. 4_! + ((4)) - (4) (4)!((4))- 44 (4)!(4)+ oscc(4) + 4 [((4)) -(4 4 4)168 (44 - 4) 4 _((4)) + 44] (4) _4!. 4+ 4!] 4(4! +4) (4 + 4) ((4)) 4 44 (44 - 4) . 4

Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.10 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 169 (4! +4)44 (4)!(4)+ 4Z(4)_4 + 4. 4_417u 44%+ C(4 ,4) 44%- _ 4. 4

%(4)!((4))+ 44171 44%- 4!. 4

(4) +(4!)4 + 4 (((4)) - 4) 4. 4

oscc(4) + 4 + .4. 4172 (44 4) - 4 44%- 4! - 4 (4) -(4) - 44((4)) + ((4))+. 4

. 4

olog(4) +4! + 4! + 4!17S (4) -((4)) + (4)4 [(4) +(4)4+4[((4)) (4) + 4 + 4 ((4)) + 4! + 44174 (44 4) - 4 (4 + 4)! - 4!4 (4)! -olog(4) - 4417S 4! + 4. 4 .4 44%- olog(4)4 [(4) -(4) 4. 4((4))! - 4! + 44 ((4) + 4)4 - ((4)) _4 oton [(4)] - 4. 4176 44 4 4 (4 +4)!4 - 4 4 + 44% - 4!44%+ ((4)) + ((4)) _4. 4_! + ((4)) + 4 C(4 ,4) + .4. 4(4) - 4! - 44 (4! (4)) - 4Z(4) 44 - ((4) 4)(4 4Z(4)) + ((4)) olog(4) + ((4))!! + 4! + 4177 (4) - _ 4. 4

%4 (4)!(4)+ ((4) 4) [((4)) - 44 - (4) Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.11 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 178 (44 4) + 4 (4 + 4)!4 - 4 ((4)) + ((4)) + 4. 444%+ _4!4] 44%- 4! + 4179 _4 oton [(4)] - 44 44%- (4 + 4) [((4)) -44 - 418u (44 4) + 4 (4 +4 + 4)!4 4. 4

%- _4!4] !4. 4 4. 4

44%- 4. 4181 _4 oton [(4)] + 44 44%- ((4)) + 4 [((4)) -44 - (4)((4) 4)+. 4

. 4

((4)) + 4! + .4. 4182 (4 + 4)!4 + 4 (44 4) + (4) (4) - 44%- 444%- 4. 4

18S _4 oton [(4)] + _4. 4

44%- ((4)) + 4 (4)!(4)+ _4!4]184 (4 +4)!4 + 4 44%- (4 4) (4 + 4) - 44%(4! 4) - (4 4) 4 (44 + 4)18S _4 oton [(4)] + 4. 4 [((4)) - 44 - 4 (4)!(4)+ ((4) 4)44%- _4. 4_ 186 (4 + 4)! + 4!4 _4 oton _44]_ + (4) 44%+ 4!. 4

44%- (4) -4 187 oton [(4)(4!)% - 44 [(4 + 4) - 44 ((4))! + 4! + 44 Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.12 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 188 (4! 4 4) - 4 44% - 4!4 4 _ . 4. 4

%+ 4]_(4)! -(4) +4Z(4) (4) - 4 - 44189 _4 oton [(4)] + 4. 4

[((4)) - 44 + 4 (4)!(4)+ (((4)) 4)(4 + 4) 4. 4

19u (4! 4 4) - 4 44%- 4. 4 44%+ . 4. 4

%(((4)) - 4) 4. 4 [4 ((4)) +. 4

. 4

191 44%- 4. 4

((4) + 4) + 4! - 4 (4 +4) - 44((4)) + ((4)) + .4. 4192 (4! 4) + (4! 4) 44%- 4 - 4 __44! _4. 4

4! (4 + 4 +4)19S ((4) + 4) +4! - 4 olog(4) +osin [(4) + _4. 4

_((4)) 4. 4

] + 4 ((4) +4)4 - (4) (4 +4) + 44194 (4! 4 4) + 4 ((4) + 4)4 -4 44%- 4 - 4(4! - 4) - (4 4)19S 44% -4. 4 _4. 4+ 4] + 4! [(4) + (4) 4. 4196 (4! 4 4) + 4 _4. 4+ 4]4 44%- 4 - 444 - 4!. 4 Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.13 4s02_0101-0200.docx SECTION II SOLUTIONS 101200 197 44% -_4. 4

((4) + 4) +4! + 4 [((4)) - 4! - 4. 4(4) - [((4)) + 44198 44% - 44 ((4) + 4)4 +4 [((4)) 4 + 4 + 4(4) ___44! + 4_ _4 oton [(4)] + 4. 4

199 44%- 44 ((4) + 4) + 4! + 4 [(4 + 4) - 4.4osin [(4) - 4+. 4

. 4

2uu (4! - 4) 4. 4 4 (4! +4! +4) _4! + 44] 4(4! 4) + (4 4) (44 4) + 4! ((4))Z(4)- 4444%+ 44% 44 - (4)!((4))! [4 ((4)) - 4. 4_(4)! - (4) +__44! _4 oton [(4)] + 4! - 4_(4) [4 + (4)] + 4 Fred E. Lusk III The Fours Game 101 to 200 04/24/1993~06/30/1998~1/25/2012 Page II.14 4s02_0101-0200.docx Solutions History 101200 # 199S 1998 2u12 1u1 1 1 4 1u2 2 2 4 1uS 1 1 4 1u4 2 2 4 1uS 2 2 7 1u6 S S S 1u7 4 4 6 1u8 4 4 7 1u9 S S S 11u 2 2 1u 111 S S S 112 2 2 4 11S S S S 114 1 1 S 11S 1 1 4 116 2 2 4 117 1 1 S 118 1 1 S 119 2 2 S 12u 2 2 S 121 S S 7 122 1 1 4 12S 2 2 4 124 2 2 7 12S 2 2 4 126 1 1 S 127 1 1 4 128 2 2 S 129 2 2 S 1Su 2 2 S 1S1 1 1 4 1S2 1 1 S 1SS 1 1 S 1S4 2 2 S 1SS 1 1 4 1S6 2 2 6 1S7 4 4 S 1S8 1 1 S 1S9 2 2 4 14u 4 4 S # 199S 1998 2u12141 1 1 7142 S S S14S 2 2 S144 S 6 1114S 2 2 S146 4 4 S147 2 2 S148 4 4 6149 S S S1Su 7 7 111S1 S S 61S2 2 2 S1SS 2 2 S1S4 1 1 41SS 2 2 S1S6 1 1 S1S7 1 1 41S8 1 1 S1S9 1 2 416u S S 4161 1 1 S162 1 1 S16S 1 1 S164 1 2 416S 1 1 S166 2 2 7167 1 1 4168 1 1 6169 2 2 S17u S S S171 1 1 4172 1 1 S17S 1 1 4174 S S S17S 2 2 6176 1 1 11177 1 1 S178 2 2 S179 1 1 S18u S S S# 199S 1998 2u12181 1 1 S182 2 2 418S 1 1 S184 1 1 S18S 1 1 4186 1 1 4187 1 1 S188 2 2 S189 1 1 419u 2 2 S191 1 1 4192 2 2 419S 1 1 S194 1 1 419S 1 1 S196 1 1 4197 1 1 4198 2 2 S199 1 1 42uu S S 12L 188 191 464 Fred E. Lusk III The Fours Game 201 to 300 04/24/1993~06/30/1998~1/25/2012 Page III.1 4s03_0201-0300.docx SECTION III SOLUTIONS 201300 2u1 44%+ 44 _(4)! + _4. 4

]4 (4)!(4) +(4! 4)4. 4

4%- 4! (4 (4)) + .4. 42u2 44%+ 4 - 4 4 _ 44%+ (4)_ _(4)! + (4) + 44%4 _(4)!(4)- 4_ ((4))!. 4

- 442uS 44% +_4. 4

[(4 + 4) -4! - 4 (4)!(4)+ olog(4) - 42u4 4 + 44% + 4 (4 + 4)!4 + 4! _(4)!(4) 4_ - (4)_(4)! + 4! 4!. 4

4 _ 44%+4]2uS 44% +4. 4 (4)!(4)+ 44% [(4 +4) - 4! - 42u6 44%+ 4 + 4 _(4)!(4) 4_ - 4 [((4)) - 4! - 442u7 [4! - (4) 4. 4

44%+ 4 - (4) [ [((4)) - 4! 442u8 44%+ 4 + 4 _(4)!(4) 4_ - 4 [((4)) - 4! + 44((4))Z(4)- 4 - 4 osin [(4) + 4+. 4

. 4

2u9 44%+ 4. 4

44%+ 4 + (4) [(4 +4) - 4! + 421u 44%+ 4. 4 (4 + 4)!(4) 4 C(4 ,4) _4. 4

(4 + 4) 4. 4 ((4))Z(4)- 4 - 4 4 44% + (4) Fred E. Lusk III The Fours Game 201 to 300 04/24/1993~06/30/1998~1/25/2012 Page III.2 4s03_0201-0300.docx SECTION III SOLUTIONS 201300 211 44 - oton _44] 44%+ 4 + (4) [(4 + 4) -4! + 4[4 ((4)) + .4. 4212 44 - 44 4! 4. 4

- 4 44% + 4!4___(4 + 4)4!- 4 ((4))Z(4)- 4 - 4 4 _ 44%+(4)_21S 4! 4. 4

- (4) 44%+(4) + (4) (4 +4)4! + (4) [(4 +4) - 4! + (4) _(4)! +(4) + 4. 4

214 4! 4. 4

- 4 ___(4 + 4)4!- 4 44% +4 + (4)(4! (4)) + C(4 ,4) . 4

. 4

%+ (4)(4) (4 + 4)4! + 421S (4! 4)-. 4

. 4

(4)!(4)+ olog(4) - (4) ((4))Z(4)- 444. 4

4%- (4) (see footnote 1) [(4 + 4) - (4 4) oton _-44] +((4) 4) 1 Theie aie at least foui methous foi cieating 22S with thiee 4s, which is the basis foi this paiticulai solution foi 21S: 22S = 4. 4

4% = ((4)) + (4)!(4) = __4. 4__4= 4. 4 oton [(4) Foi efficiency, I usually use just the fiist methou foi ueveloping complete solutions baseu on 22S foi othei integeis anu uo not show the othei thiee solutions. Similaily, when using numbeis with multiple paitial solutions (e.g. 12, S2, 64, etc.) to builu complete solutions, I geneially use only one paitial solution pei integei iathei than cieate the auuitional similai solutions. 0ften the selection of one paitial solution ovei anothei is baseu on similaiity with othei paits of the solution. Foi example, one solution foi 7S is _C(((4)), 4) - P((4), 4) = S,98S - S6u. 0f the two two4 paitial solutions I have foi S6u, I chose heie the one baseu on the Peimutation Function because the only two4 paitial solution I have foi S,98S is baseu on the ielateu Combination Function. Fred E. Lusk III The Fours Game 201 to 300 04/24/1993~06/30/1998~1/25/2012 Page III.3 4s03_0201-0300.docx SECTION III SOLUTIONS 201300 216 44%+ (4 4) ___(4 + 4 + 4)4!(4 +4)4.4

4 _44%+4_ 4 _ 44%+ 4] (4!) - ((4 + 4) 4) [(4 + 4) - _4. 4_217 (4! 4)+. 4

. 4

4. 4

4%- 4 ((4))Z(4)+ 44___(4 + 4)4!+ (4) ((4)) + (4)!(4)- 4218 ___(4 + 4)4!+ 4 44%+ 4. 4

44 - (4) -44 _olog(4) + 4. 4

]219 4. 4

4%- (4) [(4 + 4) - 4!4 ((4)) + (4)!(4)- (4) [((4)) - 4 - 4 - 422u 44 4. 4 ___(4 + 4)4!+ 4 4!+. 4

. 4

4((4)) - 4 - 4(4)221 4. 4

4%- 4 [(4 + 4) - 4. 4 [((4)) - 4 - 4 - 444 + 4%4% osin [(4) - 4 +.4. 4 C [((4)), (4) - 4(4)_4 _(4)!(4)+ 4__ + (4) Fred E. Lusk III The Fours Game 201 to 300 04/24/1993~06/30/1998~1/25/2012 Page III.4 4s03_0201-0300.docx SECTION III SOLUTIONS 201300 222 ___(4 + 4)4!+ (4) 44%+ 4! - 4 [(4 + 4) - 4. 4

_(4)! + _oton [-(4)] - ((4))Z(4)22S 4. 4

4%- 4 ((4)) + (4)!(4)- 4 4. 4

%- 44oton [-(4) + (4)(4) + 4! [(4 + 4) - 4 - 4224 4 + 44% + 4! 44%-. 4

. 4

4. 4

%- 44_(4)! +olog(4) - (4! (4)) 4 ((4 4) + 4!)22S 44%4. 4

4. 4 oton _44] (4)(4 + 4) . 4

%_4. 4_! + (4)!(4) [(4 + 4) - 4 - 4226 44%+ 4! + 4 44%+. 4

. 4

___(4 + 4)4!+ (4)4. 4

%+ 44 4. 4

4% +(4)227 4. 4

4%+ 4 ((4)) + (4)!(4)+ 4 [(4 + 4) - 4 - 4(4) +C(4 ,4)4 4. 4

%+ 44 44%+ [(4)Z(4)228 44%+ 4! + 4 44 - 4! - 4 ___((4))4!+ 4!44. 4

4%+ (4) Fred E. Lusk III The Fours Game 201 to 300 04/24/1993~06/30/1998~1/25/2012 Page III.5 4s03_0201-0300.docx SECTION III SOLUTIONS 201300 229 4. 4

4%+ 4 [(4 + 4) - 4 + 4 44 -[(4)Z(4) _4. 4

] + oscc(4) - 42Su 44%+ _ 4. 4

% [4! - (4) 4. 4 44 -4! - 4(4! (4)) - 4. 4 C(4!, 4) - 44 - 4 (4!) - C(4 + 4,4)2S1 4. 4

4%+ (4) [(4 +4) 44 _((4)) 4. 4] +((4))osin [(4) + 4 +.4. 4 (4!) - C(4 ,4) + (4) _4. 4

] + oscc _ 44]2S2 (4 + 4)4-4! 4 (olog(4) +44) 44%+ 4.42SS 4. 4

4%+ 4 [(4 + 4) + 4 - 4 _4. 4

] + oscc(4) + 42S4 44 -4! + 4 44%+ 4! + (4) _. 4

((4))] + 4!42SS 4. 4

4%+ (4) [(4 + 4) + 4 + 4 4. 4 [((4)) - (4)((4) + 4) + (4)(4) _4. 4

] + oscc(4) + 4 olog(4) + 4. 4

. 4

2S6 44 - 4! + 4 __4. 4_! 4_ - 4 _4!. 4- (4)_ 4(4!) - (4 4 4) (4!) - C(4 ,4) + (4) [((4)) +4 +44[4. 4

-(4)4 44%+ 4!. 4

. 4 _ 4!4%- (4)_(4) - 44%- ((4)) ((4))! - 4!4(4) C(4!, 4) - (4) - 4__4!4] +olog(4) - (4) Fred E. Lusk III The Fours Game 201 to 300 04/24/1993~06/30/1998~1/25/2012 Page III.6 4s03_0201-0300.docx SECTION III SOLUTIONS 201300 2S7 4. 4

%+ 4!4 [(4 + 4) + 4 + 4 (((4)) + 4!) _4. 4

_4. 4

] + oscc(4) + (4)2S8 4. 4%- 4!4 __4. 4_! 4_ -4 44 - 4. 4

2S9 (4! 4) -.4. 4 __4. 4_! 4_ - (4) [(4 + 4) + 4 + 424u 4! (4 +4 + 4) 44 - 44 4! 4. 4

. 4

_4. 4_! 44 (4 + 4) _ 4. 4

% (4 + 4)!4 . 4

(44) - ocsc(4)4 (4)4 + 4! + 4! (4) +(4! 4)4241 (4! 4) +.4. 4 __4. 4_! 4_ + (4) 44 - _4. 4_4! + 4! + 4%4% 242 (4! -4)44 __4. 4_! 4_ +4 44 - (4) - 424S 4! + 4!(. 4

)(. 4

) __4. 4

__4.4 _ [((4)) + 4 + 4 + 4244 4! + .4. 4 4 4. 4%- 4!4 __4. 4_! 4_ + 444%+ 44 44 - 4!4 _4! 4. 4] + 4olog(4) +4+. 4

. 4

Fred E. Lusk III The Fours Game 201 to 300 04/24/1993~06/30/1998~1/25/2012 Page III.7 4s03_0201-0300.docx SECTION III SOLUTIONS 201300 24S 44% -4. 4 44%+ oton _44] (44) - (4)4246 44 - 4. 4 (4). 4

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