Merry Christmas�Happy New YearLawrence E. Jones

Windham Hi^h School, WiIIitnantic, Connecticut

The days just prior to vacation are often lost to educational ob-jectives due to the restlessness that overcomes students as theirthoughts are attracted to the coming holidays. Recreational mathe-matics is perhaps in order. The copy on the facing page is both rec-reational and mathematical. With a little effort from the teacher adelightful and enlightening period can be spent deriving the requiredformulas.

Inductive reasoning can very effectively be employed to discoverthe formula for the triangular arrangement. By actual count for 2, 3,4, 5 letter triangulations, a table can be constructed.

M Y M BME YO MA BR

YOU MAR B R I

MARY iB R I A

BRIAN

r = 2 r == 3 r = 4 r = 5

T == 2 T == 4 r == 8 F==16

Students will quickly notice the doubling pattern, and many willdiscover the general formula

T == 27-1

The conjecture that a triangulation of r+1 rows can be read intwice the number of ways that r rows can be read is easily establishedas follows. Consider some particular value of r for which T (ways toread a triangular arrangement) is known. Call T==x. Let x label thespot of the last letter which can be reached x ways. Then for r+1rows, partition the diagram into triangles of r rows. Clearly y isreached only from either x and therefore must equal their sum.Hence y == 2x

0

0 0

Q 0 x

T, == x TVn == y == 2x

766

Merry Christmas�Happy New Year 767

M

M E

M E R

M E R R

MERRYM E R R Y C

M E R R Y C HM E R R Y C H R

MERRYCHR I

MERRYCHRIS

MERRYCHR I ST

MERRYCHR I STM

MERRYCHR I STMA

MERRYCHR I STMA S

HA P P Y N E W

A P P Y N E W Y

PPYNEWYEPYNEWYEA

YNEWYEAR

This diagram is pleasing to look at and innocent in appearance. How-ever, it has quite an interesting numerical biography which can besummed up in a single question: how many different ways canMERRY CHRISTMAS HAPPY NEW YEAR be read? Would youguess 25? 100? 1500? or perhaps a bold guess of 1,000,000?Merry Christmas= 2r-l where r is the number of rows. This is the

general formula for any word in a triangular pattern. In this caser==14.

(r + c - 2)!Happy New Year = ���������~ppy

(,-1)!(,-1)[

where r is the number of rows and c is the number in each row. Inthis case r== 5 and c== 8. This is the general formula for any word in arectangular pattern.The total number of ways MERRY CHRISTMAS HAPPY

NEW YEAR can be read is the product of the number of ways each

768 School Science and Mathematics

pattern can be read separately. Calculate these two values and multi-ply them and see how close you guessed.To complete the induction for T:==2r~l we observe:

1) By count that r= 1, T== 1By formula that/(l)=21-l==20== 1

2) Let r^k, the largest value of the set of values for which theformula T::=2r~l is known to be valid. Thus

f(k) = 2^

3) The (k+ l)th case requires ^+1= 2f(k) == 2(2^-1) == 2k4) By formula :/(&+1) = 2<&+l)-l== 2k

The number of ways a rectangular message can be read (R) maybe determined in the following way. Consider the word EDUCA-TION in rectangular form

E D U C A T

D U C A T I

U C A T I 0

C A T I ON

Reading always to the right or down, one solution might be

E D U

C

ATI0 N

It will be convenient to code the rectangle for easier reference.

a b c e g j

b d f h k n

c f i m p r

e h m q s t

Since there is only one starting point, a, assign a the value 1. Lettersb can be reached only one way, from a, thus assign b the value 1. Theletter d can be reached from either &, so letter d is given the value 2.The letters c can be reached only one way and are assigned the value1.

Merry Christmas�Happy New Year

1 1 1 e � �

�

1 2f �

� �

�

iy .....

e ......

769

The letters/ can be reached from c and also from d, which are valued1 and 2 respectively, so / is given the value of their sum 3. If thisanalysis is continued it is apparent that the value of each letter is thesum of the values of two letters, the one to the left and the one above.Accordingly EDUCATION can be read 56 ways as the lower rightnumber indicates.

111111

123456

1 3 6 10 15 21

1 4 10 20 35 56

If we now generalize, the first row (r== 1) is all ones. Elements ofthe second row (r== 2) have the general form c, where c is the numberof columns. Row 3 (^==3) is a sequence for which the general term is^/2(^+1). General expressions for the terms of rows 4 and 5 arec/6{c+l)(c+2) and ^/24(/;+1)(^+2)(/;+3). Since the denominators2, 6, 24 are factorial numbers for r=3, 4, 5 respectively, they mayeach be expressed as (r� 1)! Thus for r== 5, (r� 1) !=4!= 1 � 2 -3 -4= 24

column I 2 3 4 5 6 - � � c

row 1

2

3

4

5

r

1

1

1

1

1

1

2

3

4

5

1

3

6

10

15

1

4

10

20

35

1

5

15

35

70

1 ...

6 ...

21 . �

�

56 �

� �

126 �

� ’

1

c

0 ,

T^^-,(c+l)(c+2)

L’ \5

c

2-3-4c

{r-

+

1̂)!

1)

+l)(c+2)(c+3)

(c+1)^+1) ’-[c 4- fr- 2)]

Now, if the numerator and denominator are multiplied by {c-� 1)! thefinal formulation becomes

R ==(r+c-2)\

(r-l)!(c-l)!

770School Science and Mathematics

Alert students will observe that the above table is the famous tri-angle of Pascal, for which formulae are well known. A careful labelingof rows, columns, and diagonals will reveal that any entry, the cthterm of the rth row, is also the coefficient of the cth term in the ex-pansion of the (r+^�2)th power of the binomial (a-}-b). Thus

R ^ + c - 2\^

(r + c - 2) I______/r + c - 2\

\ c - 1 7c - 1 7~

!(/ + c - 2) - (^l)]\{c--~\^\(r+c-2)[

(r-l)!(^-l)!

Illustrated on the diagram above is the solution for the wordEDUCATION for which r=4 and c=6. Observe that the result 56is the coefficient of the 6th term of the 8th pov/er expansion, where6==c and 8=(r+^�2).The required calculation for the rectangular array is readily com-

Merry Christmas�Happy New Year 771

pleted by the following mnemonic device. Number the rectangle ofrequired size from upper left starting with zero. Count down the leftside, also count to the right and down the right side. Strike out thetop row. The remaining numbers in the left column are multiplied forthe denominator, and the product of the numbers in the right columnform the numerator.HAPPY NEW YEAR in the 5X 8 rectangular form would look as

follows.

0 1 2 3�4�5 6 7

1 88-9-10-11

2 97?== ������ == 3302-3-4

3 10

4 11

The original question can now be answered. How many ways canone read the message MERRY CHRISTMAS HAPPY NEWYEAR?

(5 + 8 - 2)!MCHNY = 214-1--

(5 - 1)!(8- 1)!11-10-9-8-7!

^ 2^3 . __________

2-3-4-7!

= 8192-330

== 2,703,360 ways

REFERENCESBAKST, AARON. ’^Mathematical Recreations." Mathematics Teacher^ 1952,

45:600-601.RANSOM, WILLIAM R. Thirty Projects. Portland, Maine: J. Weston Walch, 1961.

BLOOD TYPE AND FERTILITYBlood group studies may give more genetic information on a human popula-

tion than all other sources combined, say a group of Scottish scientists.From blood studies on 212 infertile men, the Scots seem to have found a rela-

tionship between blood type and rate of population growth in a community. Thelink between blood type and fertility is speculative, but blood group A seems toincrease and blood group 0 decrease among men who have few live sperm cells,report Dr. John Grieve and his co-workers at Queen’s College, Dundee.

Conversely, in India where the population is young and growing fast, onlyabout 25 percent of the people have group A blood. But in Europe where growthis considerably less, about 40 percent of the people have type-A blood.