HOMEWORK ASSIGNMENT 9

ACCELERATED PROOFS AND PROBLEM SOLVING [MATH08071]

Each problem will be marked out of 4 points.

Exercise 1 ([1, Exercise 20.2]). Let f and g be the following permutations in S7:

f =

(1 2 3 4 5 6 73 1 5 7 2 6 4

), g =

(1 2 3 4 5 6 73 1 7 6 4 5 2

).

Write down in cycle notation the permutations f , g, g2, g3, f ◦ g, (f ◦ g)−1, g−1 ◦ f−1.What is the order of f? What is the order of f ◦ g?

Solution. We have f = (1 3 5 2)(4 7), g = (1 3 7 2)(4 6 5), g2 = (1 7)(3 2)(4 5 6),g3 = (1 2 7 3), f ◦ g = (1 5 7)(2 3 4 6), (f ◦ g)−1 = (1 7 5)(2 6 4 3), and

g−1 ◦ f−1 = (f ◦ g)−1 = (1 7 5)(2 6 4 3).

The order of f is 4 by [1, Proposition 20.4]. Similarly, the order of f ◦ g is 12.

Exercise 2 ([1, Exercise 20.4]). A pack of 2n cards is shuffled by the “interlacing” methoddescribed in [1, Example 20.7] — in other words, if the original order is 1, 2, 3, 4, . . . , 2n,the new order after the shuffle is 1, n+1, 2, n+2, . . . , n, 2n. Work out how many times thisshuffle must be repeated before the cards are again in the original order in the followingcases:

(a) n = 10(b) n = 12(c) n = 14(d) n = 16(e) n = 24(f) n = 26 (i.e., a real pack of cards).

Investigate this question as far as you can for a general n — it is quite fascinating.

Solution. In other words, we must find the order of the permutation(1 2 3 4 5 6 · · · 2n− 1 2n1 n+ 1 2 n+ 2 3 n+ 3 · · · n 2n

)∈ S2n

for n = 10, 12, 14, 16, 24, 26. Let us find the order of the above permutation in the casesn = 10, 12, 14, 16, 24, 26 step by step. See Appendix for further investigation (for all nfrom 4 till 1000).

(10) The case n = 10. The first cycle is a cycle of length 18. This cycle looks like this:

(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).

The permutation is a composition of 3 cycles of lengths 1, 18, and 1. Namely, wehave the following composition of cycles:

(1)(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3)(20).

Omitting the cycles of length one, we have the following composition of cycles:

(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).

This assignment is due on Thursday 26th November 2015.

1

So, the order of the permutation is 18 by [1, Proposition 20.4].(12) The case n = 12. The first cycle is a cycle of length 11. This cycle looks like this:

(2 13 7 4 14 19 10 17 9 5 3).

The permutation is a composition of 4 cycles of lengths 1, 11, 11, and 1. Namely,we have the following composition of cycles:

(1)(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11)(24).

Omitting the cycles of length one, we have the following composition of cycles:

(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11).

So, the order of the permutation is 11 by [1, Proposition 20.4].(14) The case n = 14. The first cycle is a cycle of length 18. This cycle looks like this:

(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3).

The permutation is a composition of 5 cycles of lengths 1, 18, 6, 2, and 1. Namely,we have the following composition of cycles:

(1)(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19)(28).

Omitting the cycles of length one, we have the following composition of cycles:

(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19).

So, the order of the permutation is 18 by [1, Proposition 20.4].(16) The case n = 16. The first cycle is a cycle of length 5. This cycle looks like this:

(2 17 9 5 3).

The permutation is a composition of 8 cycles of lengths 1, 5, 5, 5, 5, 5, 5, and 1.Namely, we have the following composition of cycles:

(1)(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)(32).

Omitting the cycles of length one, we have the following composition of cycles:

(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31).

So, the order of the permutation is 5 by [1, Proposition 20.4].(24) The case n = 24. The first cycle is a cycle of length 23. This cycle looks like this:

(2 25 13 7 4 26 37 19 10 29 15 8 28 38 43 22 35 18 33 17 9 5 3).

The permutation is a composition of 4 cycles of lengths 1, 23, 23, and 1. So, theorder of the permutation is 23 by [1, Proposition 20.4].

(26) The case n = 26. The first cycle is a cycle of length 8. This cycle looks like this:

(2 27 14 33 17 9 5 3).

The permutation is a composition of 9 cycles of lengths 1, 8, 8, 8, 8, 8, 2, 8, and1. Omitting the cycles of length one, we have the following composition of cycles:

(2 27 14 33 17 9 5 3)(4 28 40 46 49 25 13 7)(6 29 15 8 30 41 21 11)

(10 31 16 34 43 22 37 19)(12 32 42 47 24 38 45 23)(18 35)(20 36 44 48 50 51 26 39).

So, the order of the permutation is 8 by [1, Proposition 20.4]. Note that 8 isrelatively small comparing to the pack size 52. I guess this is the reason why thisinterlacing trick is so popular among gamblers.

2

Exercise 3 ([1, Exercise 20.6]). This question is about the 3 × 3 version of the FifteenPizzle of Example 20.10. Starting with the configuration

1 2 34 5 67 8 �

which of the following configurations can be reached by a sequence of moves?

3 2 14 5 67 8 �

,1 � 23 4 56 7 8

,1 7 26 4 53 8 �

Solution. Arguing as in [1, Example 20.10], we see that we never can reach the configu-ration

3 2 14 5 67 8 �

,

because the corresponding permutation(1 2 3 4 5 6 7 83 2 1 4 5 6 4 8

)= (1 3)

is odd. On the other hand, we can reach the remaining two configurations

1 � 23 4 56 7 8

and1 7 26 4 53 8 �

.

Let us show this other way around. Namely, let us show how to start with any of thistwo configuration and get the initial configuration

1 2 34 5 67 8 �

.

For the first case we have

1 � 23 4 56 7 8

−→1 4 23 � 56 7 8

−→1 4 2� 3 56 7 8

−→1 4 26 3 5� 7 8

−→1 4 26 3 57 � 8

−→1 4 26 3 57 8 �

−→

−→1 4 26 3 �7 8 5

−→1 4 26 � 37 8 5

−→1 4 2� 6 37 8 5

−→1 4 27 6 3� 8 5

−→1 4 27 6 38 � 5

−→1 4 27 � 38 6 5

−→

−→1 � 27 4 38 6 5

−→1 2 �7 4 38 6 5

−→1 2 37 4 �8 6 5

−→1 2 37 4 58 6 �

−→1 2 37 4 58 � 6

−→1 2 37 4 5� 8 6

−→

−→1 2 3� 4 57 8 6

−→1 2 34 � 57 8 6

−→1 2 34 5 �7 8 6

−→1 2 34 5 67 8 �

.

For the second case we have

1 7 26 4 53 8 �

−→1 7 26 4 53 � 8

−→1 7 26 4 5� 3 8

−→1 7 2� 4 56 3 8

−→1 7 24 � 56 3 8

−→

3

−→1 7 24 3 56 � 8

−→1 7 24 3 56 8 �

−→1 7 24 3 �6 8 5

−→1 7 24 � 36 8 5

−→1 7 24 8 36 � 5

−→

−→1 7 24 8 3� 6 5

−→1 7 2� 8 34 6 5

−→1 7 28 � 34 6 5

−→1 � 28 7 34 6 5

−→1 2 �8 7 34 6 5

−→

−→1 2 38 7 �4 6 5

−→1 2 38 7 54 6 �

−→1 2 38 7 54 � 6

−→1 2 38 � 54 7 6

−→1 2 3� 8 54 7 6

−→

−→1 2 34 8 5� 7 6

−→1 2 34 8 57 � 6

−→1 2 34 � 57 8 6

−→1 2 34 5 �7 8 6

−→1 2 34 5 67 8 �

.

Now reversing the arrows, we start with the configuration

1 2 34 5 67 8 �

and get the following configurations

1 � 23 4 56 7 8

and1 7 26 4 53 8 �

.

Exercise 4 ([3, Problem 5]). Denote by α the permutation(1 2 3 4 5 6 73 7 4 1 6 5 2

)∈ S7.

(a) Express α as a product of disjoint cycles.(b) Find the order of the permutation α and find

α ◦ α ◦ α ◦ · · · ◦ α ◦ α︸ ︷︷ ︸2011 times

.

Solution. α = (1 3 4)(2 7)(5 6). Then the order of α is 6 by [1, Proposition 20.4]. Then

α ◦ α ◦ α ◦ · · · ◦ α ◦ α︸ ︷︷ ︸2011 times

= α2011 = α335×6+1 =(α6)335

α = α = (1 3 4)(2 7)(5 6).

Exercise 5 ([2, Problem 5]). Let f, g be the permutation in S7.

f =

(1 2 3 4 5 6 74 5 7 3 2 6 1

), g =

(1 2 3 4 5 6 77 6 4 5 3 2 1

).

(i) Calculate f−1 and f ◦ g.(ii) Express f as a composite of disjoint cycles.(iii) Determine whether f is an even or odd permutation.(iv) Write g as a composite of simple interchanges.

Solution. We have

f−1 =

(1 2 3 4 5 6 77 5 4 1 2 6 3

)and f ◦ g =

(1 2 3 4 5 6 71 6 3 2 7 5 4

).

We have f = (1 4 3 7)(2 5). Then f is even by [1, Proposition 20.7].4

Recall that simple interchange is just another name for a cycle of length 2. Thus,arguing as in the proof of [1, Proposition 20.6], we see that

g =

(1 2 3 4 5 6 77 6 4 5 3 2 1

)= (1 7)(2 6)(3 4 5) = (1 7)(2 6)(3 5)(3 4).

Appendix A. Interlacing cards

Let us use assumptions and notations of [1, Problem 20.4]. Let us first describe cycledecomposition for the permutation in [1, Problem 20.4] for all n from 4 to 19.

(4) The case n = 4. The permutation looks like this:(1 2 3 4 5 6 7 81 5 2 6 3 7 4 8

)The first cycle is a cycle of length 3. This cycle looks like this:

(2 5 3).

The permutation is a composition of 4 cycles of lengths 1, 3, 3, and 1. Namely, wehave the following composition of cycles:

(1)(2 5 3)(4 6 7)(8).

Omitting the cycles of length one, we have the following composition of cycles:

(2 5 3)(4 6 7).

So, the order of the permutation is 3.(5) The case n = 5. The permutation looks like this:(

1 2 3 4 5 6 7 8 9 101 6 2 7 3 8 4 9 5 10

)The first cycle is a cycle of length 6. This cycle looks like this:

(2 6 8 9 5 3).

The permutation is a composition of 4 cycles of lengths 1, 6, 2, and 1. Namely, wehave the following composition of cycles:

(1)(2 6 8 9 5 3)(4 7)(10).

Omitting the cycles of length one, we have the following composition of cycles:

(2 6 8 9 5 3)(4 7).

So, the order of the permutation is 6.(6) The case n = 6. The permutation looks like this:(

1 2 3 4 5 6 7 8 9 10 11 121 7 2 8 3 9 4 10 5 11 6 12

)The first cycle is a cycle of length 10. This cycle looks like this:

(2 7 4 8 10 11 6 9 5 3).

The permutation is a composition of 3 cycles of lengths 1, 10, and 1. Namely, wehave the following composition of cycles:

(1)(2 7 4 8 10 11 6 9 5 3)(12).

Omitting the cycles of length one, we have the following composition of cycles:

(2 7 4 8 10 11 6 9 5 3).

So, the order of the permutation is 10.5

(7) The case n = 7. The permutation looks like this:(1 2 3 4 5 6 7 8 9 10 11 12 13 141 8 2 9 3 10 4 11 5 12 6 13 7 14

)The first cycle is a cycle of length 12. This cycle looks like this:

(2 8 11 6 10 12 13 7 4 9 5 3).

The permutation is a composition of 3 cycles of lengths 1, 12, and 1. Namely, wehave the following composition of cycles:

(1)(2 8 11 6 10 12 13 7 4 9 5 3)(14).

Omitting the cycles of length one, we have the following composition of cycles:

(2 8 11 6 10 12 13 7 4 9 5 3).

So, the order of the permutation is 12.(8) The case n = 8. The permutation looks like this:(

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 161 9 2 10 3 11 4 12 5 13 6 14 7 15 8 16

)The first cycle is a cycle of length 4. This cycle looks like this:

(2 9 5 3).

The permutation is a composition of 6 cycles of lengths 1, 4, 4, 2, 4, and 1. Namely,we have the following composition of cycles:

(1)(2 9 5 3)(4 10 13 7)(6 11)(8 12 14 15)(16).

Omitting the cycles of length one, we have the following composition of cycles:

(2 9 5 3)(4 10 13 7)(6 11)(8 12 14 15).

So, the order of the permutation is 4.(9) The case n = 9. The first cycle is a cycle of length 8. This cycle looks like this:

(2 10 14 16 17 9 5 3).

The permutation is a composition of 4 cycles of lengths 1, 8, 8, and 1. Namely, wehave the following composition of cycles:

(1)(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7)(18).

Omitting the cycles of length one, we have the following composition of cycles:

(2 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7).

So, the order of the permutation is 8.(10) The case n = 10. The first cycle is a cycle of length 18. This cycle looks like this:

(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).

The permutation is a composition of 3 cycles of lengths 1, 18, and 1. Namely, wehave the following composition of cycles:

(1)(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3)(20).

Omitting the cycles of length one, we have the following composition of cycles:

(2 11 6 13 7 4 12 16 18 19 10 15 8 14 17 9 5 3).

So, the order of the permutation is 18.6

(11) The case n = 11. The first cycle is a cycle of length 6. This cycle looks like this:

(2 12 17 9 5 3).

The permutation is a composition of 7 cycles of lengths 1, 6, 3, 6, 2, 3, and 1.Namely, we have the following composition of cycles:

(1)(2 12 17 9 5 3)(4 13 7)(6 14 18 20 21 11)(8 15)(10 16 19)(22).

Omitting the cycles of length one, we have the following composition of cycles:

(2 12 17 9 5 3)(4 13 7)(6 14 18 20 21 11)(8 15)(10 16 19).

So, the order of the permutation is 6.(12) The case n = 12. The first cycle is a cycle of length 11. This cycle looks like this:

(2 13 7 4 14 19 10 17 9 5 3).

The permutation is a composition of 4 cycles of lengths 1, 11, 11, and 1. Namely,we have the following composition of cycles:

(1)(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11)(24).

Omitting the cycles of length one, we have the following composition of cycles:

(2 13 7 4 14 19 10 17 9 5 3)(6 15 8 16 20 22 23 12 18 21 11).

So, the order of the permutation is 11.(13) The case n = 13. The first cycle is a cycle of length 20. This cycle looks like this:

(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3).

The permutation is a composition of 4 cycles of lengths 1, 20, 4, and 1. Namely,we have the following composition of cycles:

(1)(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3)(6 16 21 11)(26).

Omitting the cycles of length one, we have the following composition of cycles:

(2 14 20 23 12 19 10 18 22 24 25 13 7 4 15 8 17 9 5 3)(6 16 21 11).

So, the order of the permutation is 20.(14) The case n = 14. The first cycle is a cycle of length 18. This cycle looks like this:

(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3).

The permutation is a composition of 5 cycles of lengths 1, 18, 6, 2, and 1. Namely,we have the following composition of cycles:

(1)(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19)(28).

Omitting the cycles of length one, we have the following composition of cycles:

(2 15 8 18 23 12 20 24 26 27 14 21 11 6 17 9 5 3)(4 16 22 25 13 7)(10 19).

So, the order of the permutation is 18.(15) The case n = 15. The first cycle is a cycle of length 28. This cycle looks like this:

(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3).

The permutation is a composition of 3 cycles of lengths 1, 28, and 1. Namely, wehave the following composition of cycles:

(1)(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3)(30).

Omitting the cycles of length one, we have the following composition of cycles:

(2 16 23 12 21 11 6 18 24 27 14 22 26 28 29 15 8 19 10 20 25 13 7 4 17 9 5 3).

So, the order of the permutation is 28.7

(16) The case n = 16. The first cycle is a cycle of length 5. This cycle looks like this:

(2 17 9 5 3).

The permutation is a composition of 8 cycles of lengths 1, 5, 5, 5, 5, 5, 5, and 1.Namely, we have the following composition of cycles:

(1)(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31)(32).

Omitting the cycles of length one, we have the following composition of cycles:

(2 17 9 5 3)(4 18 25 13 7)(6 19 10 21 11)(8 20 26 29 15)(12 22 27 14 23)(16 24 28 30 31).

So, the order of the permutation is 5.(17) The case n = 17. The first cycle is a cycle of length 10. This cycle looks like this:

(2 18 26 30 32 33 17 9 5 3).

The permutation is a composition of 6 cycles of lengths 1, 10, 10, 10, 2, and 1.Namely, we have the following composition of cycles:

(1)(2 18 26 30 32 33 17 9 5 3)(4 19 10 22 28 31 16 25 13 7)(6 20 27 14 24 29 15 8 21 11)(12 23)(34).

Omitting the cycles of length one, we have the following composition of cycles:

(2 18 26 30 32 33 17 9 5 3)(4 19 10 22 28 31 16 25 13 7)(6 20 27 14 24 29 15 8 21 11)(12 23).

So, the order of the permutation is 10.(18) The case n = 18. The first cycle is a cycle of length 12. This cycle looks like this:

(2 19 10 23 12 24 30 33 17 9 5 3).

The permutation is a composition of 7 cycles of lengths 1, 12, 12, 3, 4, 3, and 1.Namely, we have the following composition of cycles:

(1)(2 19 10 23 12 24 30 33 17 9 5 3)(4 20 28 32 34 35 18 27 14 25 13 7)(6 21 11)(8 22 29 15)(16 26 31)(36).

Omitting the cycles of length one, we have the following composition of cycles:

(2 19 10 23 12 24 30 33 17 9 5 3)(4 20 28 32 34 35 18 27 14 25 13 7)(6 21 11)(8 22 29 15)(16 26 31).

So, the order of the permutation is 12.(19) The case n = 19. The first cycle is a cycle of length 36. This cycle looks like this:

(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3).

The permutation is a composition of 3 cycles of lengths 1, 36, and 1. Namely, wehave the following composition of cycles:

(1)(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3)(38).

Omitting the cycles of length one, we have the following composition of cycles:

(2 20 29 15 8 23 12 25 13 7 4 21 11 6 22 30 34 36 37 19 10 24 31 16 27 14 26 32 35 18 28 33 17 9 5 3).

So, the order of the permutation is 36.

We can continue in the same fashion for a very long time (keeping in mind that most ofthis text is written by a computer program). But it would be much better to understandthe rule that stands behind this. Unfortunately, I did not find this rule (this does notmean the rule does not exists and this does not mean that it is hard to find the rule).Nevertheless, I investigated a lot of cases (using a simple program in Java). Let mesummarize my investigations in a table whose first column contains n for all n from 4 till1000, the second column contains 2n (the number of interlacing cards), the third columncontains the cycle shape of the permutation like in [1, Example 20.4] and [1, Example 20.5](but I put the cycle shape in the ascending order contrary to the book, since it looks morenatural to me), and final forth column contains the order of our permutation (what weare looking for!).

8

n 2n cycle-shape order

4 8 (12, 32) 3

5 10 (12, 2, 6) 6

6 12 (12, 10) 10

7 14 (12, 12) 12

8 16 (12, 2, 43) 4

9 18 (12, 82) 8

10 20 (12, 18) 18

11 22 (12, 2, 32, 62) 6

12 24 (12, 112) 11

13 26 (12, 4, 20) 20

14 28 (12, 2, 6, 18) 18

15 30 (12, 28) 28

16 32 (12, 56) 5

17 34 (12, 2, 103) 10

18 36 (12, 32, 4, 122) 12

19 38 (12, 36) 36

20 40 (12, 2, 123) 12

21 42 (12, 202) 20

22 44 (12, 143) 14

23 46 (12, 2, 43, 6, 122) 12

24 48 (12, 232) 23

25 50 (12, 32, 212) 21

26 52 (12, 2, 86) 8

27 54 (12, 52) 52

28 56 (12, 4, 10, 202) 20

29 58 (12, 2, 183) 18

30 60 (12, 58) 58

31 62 (12, 60) 60

32 64 (12, 2, 32, 69) 6

33 66 (12, 4, 125) 12

34 68 (12, 66) 66

9

35 70 (12, 2, 112, 222) 22

36 72 (12, 352) 35

37 74 (12, 98) 9

38 76 (12, 2, 43, 203) 20

39 78 (12, 32, 10, 302) 30

40 80 (12, 392) 39

41 82 (12, 2, 6, 18, 54) 54

42 84 (12, 82) 82

43 86 (12, 4, 810) 8

44 88 (12, 2, 283) 28

45 90 (12, 118) 11

46 92 (12, 32, 127) 12

47 94 (12, 2, 56, 106) 10

48 96 (12, 4, 18, 362) 36

49 98 (12, 482) 48

50 100 (12, 2, 6, 103, 302) 30

51 102 (12, 100) 100

52 104 (12, 512) 51

53 106 (12, 2, 32, 43, 62, 126) 12

54 108 (12, 106) 106

55 110 (12, 363) 36

56 112 (12, 2, 363) 36

57 114 (12, 284) 28

58 116 (12, 4, 112, 442) 44

59 118 (12, 2, 6, 129) 12

60 120 (12, 32, 82, 244) 24

61 122 (12, 10, 110) 110

62 124 (12, 2, 206) 20

63 126 (12, 4, 20, 100) 100

64 128 (12, 718) 7

65 130 (12, 2, 149) 14

66 132 (12, 130) 130

67 134 (12, 32, 187) 18

10

68 136 (12, 2, 43, 6, 122, 18, 362) 36

69 138 (12, 682) 68

70 140 (12, 138) 138

71 142 (12, 2, 232, 462) 46

72 144 (12, 10, 12, 602) 60

73 146 (12, 4, 285) 28

74 148 (12, 2, 32, 62, 212, 422) 42

75 150 (12, 148) 148

76 152 (12, 1510) 15

77 154 (12, 2, 6, 86, 244) 24

78 156 (12, 4, 56, 206) 20

79 158 (12, 523) 52

80 160 (12, 2, 523) 52

81 162 (12, 32, 112, 334) 33

82 164 (12, 162) 162

83 166 (12, 2, 43, 103, 206) 20

84 168 (12, 832) 83

85 170 (12, 12, 156) 156

86 172 (12, 2, 6, 189) 18

87 174 (12, 172) 172

88 176 (12, 32, 4, 122, 20, 602) 60

89 178 (12, 2, 583) 58

90 180 (12, 178) 178

91 182 (12, 180) 180

92 184 (12, 2, 603) 60

93 186 (12, 4, 365) 36

94 188 (12, 82, 10, 404) 40

95 190 (12, 2, 32, 69, 187) 18

96 192 (12, 952) 95

97 194 (12, 962) 96

98 196 (12, 2, 43, 1215) 12

99 198 (12, 196) 196

100 200 (12, 992) 99

11

101 202 (12, 2, 663) 66

102 204 (12, 32, 28, 842) 84

103 206 (12, 4, 2010) 20

104 208 (12, 2, 6, 112, 222, 662) 66

105 210 (12, 10, 18, 902) 90

106 212 (12, 210) 210

107 214 (12, 2, 352, 702) 70

108 216 (12, 4, 143, 286) 28

109 218 (12, 32, 56, 1512) 15

110 220 (12, 2, 98, 188) 18

111 222 (12, 82, 12, 248) 24

112 224 (12, 376) 37

113 226 (12, 2, 43, 6, 122, 203, 602) 60

114 228 (12, 226) 226

115 230 (12, 763) 76

116 232 (12, 2, 32, 62, 103, 306) 30

117 234 (12, 298) 29

118 236 (12, 4, 232, 922) 92

119 238 (12, 2, 392, 782) 78

120 240 (12, 1192) 119

121 242 (12, 2410) 24

122 244 (12, 2, 6, 18, 54, 162) 162

123 246 (12, 32, 4, 122, 212, 842) 84

124 248 (12, 12, 18, 366) 36

125 250 (12, 2, 823) 82

126 252 (12, 505) 50

127 254 (12, 10, 112, 1102) 110

128 256 (12, 2, 43, 830) 8

129 258 (12, 1616) 16

130 260 (12, 32, 367) 36

131 262 (12, 2, 6, 283, 842) 84

132 264 (12, 1312) 131

133 266 (12, 4, 525) 52

12

134 268 (12, 2, 118, 228) 22

135 270 (12, 268) 268

136 272 (12, 1352) 135

137 274 (12, 2, 32, 62, 1221) 12

138 276 (12, 4, 10, 2013) 20

139 278 (12, 923) 92

140 280 (12, 2, 56, 6, 106, 306) 30

141 282 (12, 704) 70

142 284 (12, 943) 94

143 286 (12, 2, 43, 183, 366) 36

144 288 (12, 32, 202, 604) 60

145 290 (12, 82, 1362) 136

146 292 (12, 2, 486) 48

147 294 (12, 292) 292

148 296 (12, 4, 58, 1162) 116

149 298 (12, 2, 6, 103, 18, 302, 902) 90

150 300 (12, 112, 12, 1322) 132

151 302 (12, 32, 143, 426) 42

152 304 (12, 2, 1003) 100

153 306 (12, 4, 605) 60

154 308 (12, 1023) 102

155 310 (12, 2, 512, 1022) 102

156 312 (12, 1552) 155

157 314 (12, 1562) 156

158 316 (12, 2, 32, 43, 69, 1220) 12

159 318 (12, 316) 316

160 320 (12, 10, 28, 1402) 140

161 322 (12, 2, 1063) 106

162 324 (12, 82, 18, 724) 72

163 326 (12, 4, 125, 20, 604) 60

164 328 (12, 2, 369) 36

165 330 (12, 32, 232, 694) 69

166 332 (12, 3011) 30

13

167 334 (12, 2, 6, 369) 36

168 336 (12, 4, 66, 1322) 132

169 338 (12, 2116) 21

170 340 (12, 2, 2812) 28

171 342 (12, 56, 1031) 10

172 344 (12, 32, 212, 1472) 147

173 346 (12, 2, 43, 112, 222, 446) 44

174 348 (12, 346) 346

175 350 (12, 348) 348

176 352 (12, 2, 6, 129, 18, 366) 36

177 354 (12, 884) 88

178 356 (12, 4, 352, 1402) 140

179 358 (12, 2, 32, 62, 86, 2412) 24

180 360 (12, 1792) 179

181 362 (12, 18, 342) 342

182 364 (12, 2, 103, 1103) 110

183 366 (12, 4, 98, 368) 36

184 368 (12, 1832) 183

185 370 (12, 2, 6, 206, 604) 60

186 372 (12, 32, 52, 1562) 156

187 374 (12, 372) 372

188 376 (12, 2, 43, 203, 1003) 100

189 378 (12, 12, 28, 844) 84

190 380 (12, 378) 378

191 382 (12, 2, 718, 1418) 14

192 384 (12, 1912) 191

193 386 (12, 32, 4, 10, 122, 202, 302, 604) 60

194 388 (12, 2, 6, 149, 426) 42

195 390 (12, 388) 388

196 392 (12, 82, 112, 884) 88

197 394 (12, 2, 1303) 130

198 396 (12, 4, 392, 1562) 156

199 398 (12, 449) 44

14

200 400 (12, 2, 32, 62, 1821) 18

201 402 (12, 2002) 200

202 404 (12, 56, 12, 606) 60

203 406 (12, 2, 43, 6, 122, 18, 362, 54, 1082) 108

204 408 (12, 10, 36, 1802) 180

205 410 (12, 2042) 204

206 412 (12, 2, 686) 68

207 414 (12, 32, 58, 1742) 174

208 416 (12, 4, 82, 1642) 164

209 418 (12, 2, 1383) 138

210 420 (12, 418) 418

211 422 (12, 420) 420

212 424 (12, 2, 6, 232, 462, 1382) 138

213 426 (12, 4, 810, 20, 408) 40

214 428 (12, 32, 607) 60

215 430 (12, 2, 103, 123, 606) 60

216 432 (12, 4310) 43

217 434 (12, 726) 72

218 436 (12, 2, 43, 2815) 28

219 438 (12, 112, 18, 1982) 198

220 440 (12, 736) 73

221 442 (12, 2, 32, 69, 212, 428) 42

222 444 (12, 442) 442

223 446 (12, 4, 118, 448) 44

224 448 (12, 2, 1483) 148

225 450 (12, 2242) 224

226 452 (12, 10, 2022) 20

227 454 (12, 2, 1510, 3010) 30

228 456 (12, 32, 4, 1237) 12

229 458 (12, 766) 76

230 460 (12, 2, 6, 86, 18, 244, 724) 72

231 462 (12, 460) 460

232 464 (12, 2312) 231

15

233 466 (12, 2, 43, 56, 106, 2018) 20

234 468 (12, 466) 466

235 470 (12, 32, 667) 66

236 472 (12, 2, 529) 52

237 474 (12, 10, 143, 706) 70

238 476 (12, 4, 18, 20, 362, 1802) 180

239 478 (12, 2, 6, 523, 1562) 156

240 480 (12, 2392) 239

241 482 (12, 12, 3613) 36

242 484 (12, 2, 32, 62, 112, 222, 334, 664) 66

243 486 (12, 4, 4810) 48

244 488 (12, 2432) 243

245 490 (12, 2, 1623) 162

246 492 (12, 490) 490

247 494 (12, 82, 28, 568) 56

248 496 (12, 2, 43, 6, 103, 122, 206, 302, 604) 60

249 498 (12, 32, 352, 1054) 105

250 500 (12, 1663) 166

251 502 (12, 2, 832, 1662) 166

252 504 (12, 2512) 251

253 506 (12, 4, 1005) 100

254 508 (12, 2, 123, 1563) 156

255 510 (12, 508) 508

256 512 (12, 32, 956) 9

257 514 (12, 2, 6, 1828) 18

258 516 (12, 4, 512, 2042) 204

259 518 (12, 10, 232, 2302) 230

260 520 (12, 2, 1723) 172

261 522 (12, 2602) 260

262 524 (12, 522) 522

263 526 (12, 2, 32, 43, 62, 126, 203, 606) 60

264 528 (12, 56, 82, 4012) 40

265 530 (12, 112, 2532) 253

16

266 532 (12, 2, 6, 583, 1742) 174

267 534 (12, 12, 202, 608) 60

268 536 (12, 4, 106, 2122) 212

269 538 (12, 2, 1783) 178

270 540 (12, 32, 10, 212, 302, 2102) 210

271 542 (12, 540) 540

272 544 (12, 2, 1803) 180

273 546 (12, 4, 3615) 36

274 548 (12, 546) 546

275 550 (12, 2, 6, 609) 60

276 552 (12, 18, 28, 2522) 252

277 554 (12, 32, 3914) 39

278 556 (12, 2, 43, 3615) 36

279 558 (12, 556) 556

280 560 (12, 12, 143, 846) 84

281 562 (12, 2, 86, 103, 4012) 40

282 564 (12, 562) 562

283 566 (12, 4, 2820) 28

284 568 (12, 2, 32, 69, 187, 547) 54

285 570 (12, 2842) 284

286 572 (12, 1145) 114

287 574 (12, 2, 952, 1902) 190

288 576 (12, 4, 112, 20, 442, 2202) 220

289 578 (12, 1444) 144

290 580 (12, 2, 966) 96

291 582 (12, 32, 82, 2462) 246

292 584 (12, 10, 52, 2602) 260

293 586 (12, 2, 43, 6, 1247) 12

294 588 (12, 586) 586

295 590 (12, 56, 18, 906) 90

296 592 (12, 2, 1963) 196

297 594 (12, 1484) 148

298 596 (12, 32, 4, 810, 122, 2420) 24

17

299 598 (12, 2, 992, 1982) 198

300 600 (12, 2992) 299

301 602 (12, 2524) 25

302 604 (12, 2, 6, 669) 66

303 606 (12, 4, 10, 202, 110, 2202) 220

304 608 (12, 3032) 303

305 610 (12, 2, 32, 62, 283, 846) 84

306 612 (12, 12, 232, 2762) 276

307 614 (12, 612) 612

308 616 (12, 2, 43, 2030) 20

309 618 (12, 1544) 154

310 620 (12, 618) 618

311 622 (12, 2, 6, 112, 18, 222, 662, 1982) 198

312 624 (12, 32, 118, 3316) 33

313 626 (12, 4, 20, 100, 500) 500

314 628 (12, 2, 103, 183, 906) 90

315 630 (12, 82, 36, 728) 72

316 632 (12, 4514) 45

317 634 (12, 2, 2103) 210

318 636 (12, 4, 718, 2818) 28

319 638 (12, 32, 127, 212, 846) 84

320 640 (12, 2, 6, 352, 702, 2102) 210

321 642 (12, 6410) 64

322 644 (12, 2143) 214

323 646 (12, 2, 43, 149, 2818) 28

324 648 (12, 3232) 323

325 650 (12, 10, 58, 2902) 290

326 652 (12, 2, 32, 56, 62, 106, 1512, 3012) 30

327 654 (12, 652) 652

328 656 (12, 4, 130, 2602) 260

329 658 (12, 2, 6, 98, 1832) 18

330 660 (12, 658) 658

331 662 (12, 660) 660

18

332 664 (12, 2, 86, 123, 2424) 24

333 666 (12, 32, 4, 122, 187, 3614) 36

334 668 (12, 112, 28, 3082) 308

335 670 (12, 2, 376, 746) 74

336 672 (12, 10, 6011) 60

337 674 (12, 4814) 48

338 676 (12, 2, 43, 6, 122, 18, 203, 362, 602, 1802) 180

339 678 (12, 676) 676

340 680 (12, 32, 4814) 48

341 682 (12, 2, 2263) 226

342 684 (12, 2231) 22

343 686 (12, 4, 6810) 68

344 688 (12, 2, 769) 76

345 690 (12, 12, 52, 1564) 156

346 692 (12, 2303) 230

347 694 (12, 2, 32, 69, 103, 3020) 30

348 696 (12, 4, 138, 2762) 276

349 698 (12, 82, 202, 4016) 40

350 700 (12, 2, 298, 588) 58

351 702 (12, 700) 700

352 704 (12, 18, 3619) 36

353 706 (12, 2, 43, 232, 462, 926) 92

354 708 (12, 32, 100, 3002) 300

355 710 (12, 708) 708

356 712 (12, 2, 6, 392, 788) 78

357 714 (12, 56, 112, 5512) 55

358 716 (12, 4, 10, 125, 202, 6010) 60

359 718 (12, 2, 1192, 2382) 238

360 720 (12, 3592) 359

361 722 (12, 32, 5114) 51

362 724 (12, 2, 2430) 24

363 726 (12, 4, 20, 285, 1404) 140

364 728 (12, 1216) 121

19

365 730 (12, 2, 6, 18, 54, 162, 486) 486

366 732 (12, 82, 143, 5612) 56

367 734 (12, 2443) 244

368 736 (12, 2, 32, 43, 62, 126, 212, 422, 846) 84

369 738 (12, 10, 66, 3302) 330

370 740 (12, 2463) 246

371 742 (12, 2, 123, 183, 3618) 36

372 744 (12, 3712) 371

373 746 (12, 4, 1485) 148

374 748 (12, 2, 6, 823, 2462) 246

375 750 (12, 32, 106, 3182) 318

376 752 (12, 3752) 375

377 754 (12, 2, 5015) 50

378 756 (12, 4, 1510, 6010) 60

379 758 (12, 756) 756

380 760 (12, 2, 103, 112, 222, 1106) 110

381 762 (12, 3802) 380

382 764 (12, 32, 3621) 36

383 766 (12, 2, 43, 6, 830, 122, 2420) 24

384 768 (12, 12, 58, 3482) 348

385 770 (12, 3842) 384

386 772 (12, 2, 1648) 16

387 774 (12, 772) 772

388 776 (12, 4, 56, 2037) 20

389 778 (12, 2, 32, 62, 3621) 36

390 780 (12, 18, 202, 1804) 180

391 782 (12, 10, 352, 7010) 70

392 784 (12, 2, 6, 18, 283, 842, 2522) 252

393 786 (12, 4, 5215) 52

394 788 (12, 786) 786

395 790 (12, 2, 1312, 2622) 262

396 792 (12, 32, 284, 848) 84

397 794 (12, 12, 6013) 60

20

398 796 (12, 2, 43, 5215) 52

399 798 (12, 796) 796

400 800 (12, 82, 232, 1844) 184

401 802 (12, 2, 6, 118, 228, 668) 66

402 804 (12, 98, 10, 908) 90

403 806 (12, 32, 4, 112, 122, 334, 442, 1324) 132

404 808 (12, 2, 2683) 268

405 810 (12, 4042) 404

406 812 (12, 2703) 270

407 814 (12, 2, 1352, 2702) 270

408 816 (12, 4, 162, 3242) 324

409 818 (12, 143, 18, 1266) 126

410 820 (12, 2, 32, 69, 1263) 12

411 822 (12, 820) 820

412 824 (12, 4112) 411

413 826 (12, 2, 43, 103, 2039) 20

414 828 (12, 826) 826

415 830 (12, 828) 828

416 832 (12, 2, 929) 92

417 834 (12, 32, 82, 212, 244, 1684) 168

418 836 (12, 4, 832, 3322) 332

419 838 (12, 2, 56, 6, 106, 18, 306, 906) 90

420 840 (12, 4192) 419

421 842 (12, 28, 812) 812

422 844 (12, 2, 7012) 70

423 846 (12, 4, 125, 1565) 156

424 848 (12, 32, 10, 302, 110, 3302) 330

425 850 (12, 2, 949) 94

426 852 (12, 112, 36, 3962) 396

427 854 (12, 852) 852

428 856 (12, 2, 43, 6, 122, 189, 3618) 36

429 858 (12, 4282) 428

430 860 (12, 858) 858

21

431 862 (12, 2, 32, 62, 206, 6012) 60

432 864 (12, 4312) 431

433 866 (12, 4, 1725) 172

434 868 (12, 2, 86, 1366) 136

435 870 (12, 10, 392, 3902) 390

436 872 (12, 12, 66, 1326) 132

437 874 (12, 2, 6, 4818) 48

438 876 (12, 32, 4, 122, 20, 602, 100, 3002) 300

439 878 (12, 876) 876

440 880 (12, 2, 2923) 292

441 882 (12, 5516) 55

442 884 (12, 882) 882

443 886 (12, 2, 43, 583, 1166) 116

444 888 (12, 4432) 443

445 890 (12, 32, 718, 2136) 21

446 892 (12, 2, 6, 103, 18, 302, 54, 902, 2702) 270

447 894 (12, 18, 232, 4142) 414

448 896 (12, 4, 178, 3562) 356

449 898 (12, 2, 112, 123, 222, 1326) 132

450 900 (12, 56, 28, 1406) 140

451 902 (12, 82, 52, 1048) 104

452 904 (12, 2, 32, 62, 149, 4218) 42

453 906 (12, 4, 1805) 180

454 908 (12, 906) 906

455 910 (12, 2, 6, 1003, 3002) 300

456 912 (12, 9110) 91

457 914 (12, 10, 82, 4102) 410

458 916 (12, 2, 43, 6015) 60

459 918 (12, 32, 130, 3902) 390

460 920 (12, 1536) 153

461 922 (12, 2, 1029) 102

462 924 (12, 12, 352, 4202) 420

463 926 (12, 4, 20, 365, 1804) 180

22

464 928 (12, 2, 6, 512, 1028) 102

465 930 (12, 4642) 464

466 932 (12, 32, 187, 212, 1266) 126

467 934 (12, 2, 1552, 3102) 310

468 936 (12, 4, 810, 10, 202, 4020) 40

469 938 (12, 1178) 117

470 940 (12, 2, 1566) 156

471 942 (12, 940) 940

472 944 (12, 112, 202, 2204) 220

473 946 (12, 2, 32, 43, 69, 1220, 187, 3614) 36

474 948 (12, 946) 946

475 950 (12, 98, 12, 3624) 36

476 952 (12, 2, 3163) 316

477 954 (12, 6814) 68

478 956 (12, 4, 952, 3802) 380

479 958 (12, 2, 103, 283, 1406) 140

480 960 (12, 32, 682, 2044) 204

481 962 (12, 56, 1556) 155

482 964 (12, 2, 6, 1063, 3182) 318

483 966 (12, 4, 9610) 96

484 968 (12, 4832) 483

485 970 (12, 2, 86, 183, 7212) 72

486 972 (12, 1945) 194

487 974 (12, 32, 1387) 138

488 976 (12, 2, 43, 1215, 203, 6012) 60

489 978 (12, 4882) 488

490 980 (12, 10, 118, 1108) 110

491 982 (12, 2, 6, 3627) 36

492 984 (12, 4912) 491

493 986 (12, 4, 1965) 196

494 988 (12, 2, 32, 62, 232, 462, 694, 1384) 138

495 990 (12, 112, 143, 1546) 154

496 992 (12, 4952) 495

23

497 994 (12, 2, 3033) 30

498 996 (12, 4, 992, 3962) 396

499 998 (12, 3323) 332

500 1000 (12, 2, 6, 18, 3627) 36

501 1002 (12, 32, 10, 127, 302, 6014) 60

502 1004 (12, 82, 58, 2324) 232

503 1006 (12, 2, 43, 663, 1326) 132

504 1008 (12, 18, 52, 4682) 468

505 1010 (12, 5042) 504

506 1012 (12, 2, 2116, 4216) 42

507 1014 (12, 9211) 92

508 1016 (12, 32, 4, 122, 285, 8410) 84

509 1018 (12, 2, 6, 2812, 848) 84

510 1020 (12, 1018) 1018

511 1022 (12, 3403) 340

512 1024 (12, 2, 56, 1099) 10

513 1026 (12, 4, 2051) 20

514 1028 (12, 12, 392, 1566) 156

515 1030 (12, 2, 32, 62, 212, 422, 1472, 2942) 294

516 1032 (12, 5152) 515

517 1034 (12, 2584) 258

518 1036 (12, 2, 43, 6, 112, 122, 222, 446, 662, 1324) 132

519 1038 (12, 82, 60, 1208) 120

520 1040 (12, 5192) 519

521 1042 (12, 2, 3463) 346

522 1044 (12, 32, 148, 4442) 444

523 1046 (12, 4, 10, 18, 202, 362, 902, 1804) 180

524 1048 (12, 2, 3483) 348

525 1050 (12, 2624) 262

526 1052 (12, 3503) 350

527 1054 (12, 2, 6, 129, 18, 366, 54, 1086) 108

528 1056 (12, 4, 210, 4202) 420

529 1058 (12, 32, 1570) 15

24

530 1060 (12, 2, 8812) 88

531 1062 (12, 1060) 1060

532 1064 (12, 5312) 531

533 1066 (12, 2, 43, 352, 702, 1406) 140

534 1068 (12, 10, 482, 2404) 240

535 1070 (12, 3563) 356

536 1072 (12, 2, 32, 69, 86, 2440) 24

537 1074 (12, 28, 36, 2524) 252

538 1076 (12, 4, 143, 20, 286, 1406) 140

539 1078 (12, 2, 1792, 3582) 358

540 1080 (12, 12, 82, 4922) 492

541 1082 (12, 112, 232, 2534) 253

542 1084 (12, 2, 183, 3423) 342

543 1086 (12, 32, 4, 56, 122, 1512, 206, 6012) 60

544 1088 (12, 5432) 543

545 1090 (12, 2, 6, 103, 302, 1103, 3302) 330

546 1092 (12, 1090) 1090

547 1094 (12, 3643) 364

548 1096 (12, 2, 43, 98, 188, 3624) 36

549 1098 (12, 2744) 274

550 1100 (12, 32, 523, 1566) 156

551 1102 (12, 2, 1832, 3662) 366

552 1104 (12, 2938) 29

553 1106 (12, 4, 810, 125, 2440) 24

554 1108 (12, 2, 6, 18, 206, 604, 1804) 180

555 1110 (12, 1108) 1108

556 1112 (12, 10, 10011) 100

557 1114 (12, 2, 32, 62, 523, 1566) 156

558 1116 (12, 4, 376, 1486) 148

559 1118 (12, 1116) 1116

560 1120 (12, 2, 3723) 372

561 1122 (12, 18, 58, 5222) 522

562 1124 (12, 1122) 1122

25

563 1126 (12, 2, 43, 6, 122, 203, 602, 1003, 3002) 300

564 1128 (12, 32, 112, 212, 334, 2314) 231

565 1130 (12, 5642) 564

566 1132 (12, 2, 123, 283, 8412) 84

567 1134 (12, 10, 512, 5102) 510

568 1136 (12, 4, 226, 4522) 452

569 1138 (12, 2, 3783) 378

570 1140 (12, 82, 66, 2644) 264

571 1142 (12, 32, 1627) 162

572 1144 (12, 2, 6, 718, 1418, 4218) 42

573 1146 (12, 4, 7615) 76

574 1148 (12, 56, 36, 1806) 180

575 1150 (12, 2, 1912, 3822) 382

576 1152 (12, 5752) 575

577 1154 (12, 2884) 288

578 1156 (12, 2, 32, 43, 62, 103, 126, 206, 306, 6012) 60

579 1158 (12, 118, 12, 1328) 132

580 1160 (12, 18, 60, 1806) 180

581 1162 (12, 2, 6, 149, 18, 426, 1266) 126

582 1164 (12, 1667) 166

583 1166 (12, 4, 298, 1168) 116

584 1168 (12, 2, 3883) 388

585 1170 (12, 32, 832, 2494) 249

586 1172 (12, 1170) 1170

587 1174 (12, 2, 86, 112, 222, 8812) 88

588 1176 (12, 4, 20, 232, 922, 4602) 460

589 1178 (12, 10, 106, 5302) 530

590 1180 (12, 2, 6, 1303, 3902) 390

591 1182 (12, 2365) 236

592 1184 (12, 32, 127, 1567) 156

593 1186 (12, 2, 43, 392, 782, 1566) 156

594 1188 (12, 1186) 1186

595 1190 (12, 202, 28, 1408) 140

26

596 1192 (12, 2, 4427) 44

597 1194 (12, 2984) 298

598 1196 (12, 4, 1192, 4762) 476

599 1198 (12, 2, 32, 69, 1863) 18

600 1200 (12, 10, 363, 1806) 180

601 1202 (12, 3004) 300

602 1204 (12, 2, 2006) 200

603 1206 (12, 4, 2450) 24

604 1208 (12, 82, 352, 2804) 280

605 1210 (12, 2, 56, 106, 123, 6018) 60

606 1212 (12, 32, 172, 5162) 516

607 1214 (12, 1212) 1212

608 1216 (12, 2, 43, 6, 122, 18, 362, 54, 1082, 162, 3242) 324

609 1218 (12, 1528) 152

610 1220 (12, 112, 52, 5722) 572

611 1222 (12, 2, 103, 363, 1806) 180

612 1224 (12, 6112) 611

613 1226 (12, 32, 4, 122, 20, 212, 602, 842, 4202) 420

614 1228 (12, 2, 2046) 204

615 1230 (12, 1228) 1228

616 1232 (12, 6152) 615

617 1234 (12, 2, 6, 686, 2044) 204

618 1236 (12, 4, 125, 18, 3632) 36

619 1238 (12, 1236) 1236

620 1240 (12, 2, 32, 62, 583, 1746) 174

621 1242 (12, 82, 98, 7216) 72

622 1244 (12, 10, 284, 1408) 140

623 1246 (12, 2, 43, 823, 1646) 164

624 1248 (12, 143, 2843) 28

625 1250 (12, 1568) 156

626 1252 (12, 2, 6, 1389) 138

627 1254 (12, 32, 178, 5342) 534

628 1256 (12, 4, 505, 10010) 100

27

629 1258 (12, 2, 4183) 418

630 1260 (12, 1258) 1258

631 1262 (12, 12, 4826) 48

632 1264 (12, 2, 4203) 420

633 1266 (12, 4, 10, 112, 202, 442, 1102, 2204) 220

634 1268 (12, 32, 1807) 180

635 1270 (12, 2, 6, 18, 232, 462, 1382, 4142) 414

636 1272 (12, 56, 2062) 20

637 1274 (12, 18, 66, 1986) 198

638 1276 (12, 2, 43, 830, 203, 4024) 40

639 1278 (12, 1276) 1276

640 1280 (12, 6392) 639

641 1282 (12, 2, 32, 62, 6021) 60

642 1284 (12, 1282) 1282

643 1286 (12, 4, 1680) 16

644 1288 (12, 2, 6, 103, 129, 302, 6018) 60

645 1290 (12, 1618) 161

646 1292 (12, 1290) 1290

647 1294 (12, 2, 4310, 8610) 86

648 1296 (12, 32, 4, 122, 3635) 36

649 1298 (12, 6482) 648

650 1300 (12, 2, 7218) 72

651 1302 (12, 1300) 1300

652 1304 (12, 6512) 651

653 1306 (12, 2, 43, 6, 122, 2815, 8410) 84

654 1308 (12, 1306) 1306

655 1310 (12, 32, 82, 10, 244, 302, 404, 1208) 120

656 1312 (12, 2, 112, 183, 222, 1986) 198

657 1314 (12, 12, 100, 3004) 300

658 1316 (12, 4, 1312, 5242) 524

659 1318 (12, 2, 736, 1466) 146

660 1320 (12, 6592) 659

661 1322 (12, 6022) 60

28

662 1324 (12, 2, 32, 69, 187, 212, 428, 1266) 126

663 1326 (12, 4, 20, 525, 2604) 260

664 1328 (12, 2216) 221

665 1330 (12, 2, 4423) 442

666 1332 (12, 10, 110, 1210) 1210

667 1334 (12, 56, 143, 7018) 70

668 1336 (12, 2, 43, 118, 228, 4424) 44

669 1338 (12, 32, 952, 2854) 285

670 1340 (12, 12, 512, 2046) 204

671 1342 (12, 2, 6, 1483, 4442) 444

672 1344 (12, 82, 392, 3124) 312

673 1346 (12, 4, 2685) 268

674 1348 (12, 2, 2246) 224

675 1350 (12, 18, 352, 6302) 630

676 1352 (12, 32, 9614) 96

677 1354 (12, 2, 103, 2066) 20

678 1356 (12, 4, 1352, 5402) 540

679 1358 (12, 112, 58, 6382) 638

680 1360 (12, 2, 6, 1510, 3040) 30

681 1362 (12, 6802) 680

682 1364 (12, 232, 28, 6442) 644

683 1366 (12, 2, 32, 43, 62, 12111) 12

684 1368 (12, 6832) 683

685 1370 (12, 36, 1332) 1332

686 1372 (12, 2, 7618) 76

687 1374 (12, 1372) 1372

688 1376 (12, 4, 10, 2013, 10011) 100

689 1378 (12, 2, 6, 86, 18, 244, 54, 724, 2164) 216

690 1380 (12, 32, 196, 5882) 588

691 1382 (12, 1380) 1380

692 1384 (12, 2, 4603) 460

693 1386 (12, 4, 9215) 92

694 1388 (12, 98, 1873) 18

29

695 1390 (12, 2, 2312, 4622) 462

696 1392 (12, 12, 106, 6362) 636

697 1394 (12, 32, 9914) 99

698 1396 (12, 2, 43, 56, 6, 106, 122, 2018, 306, 6012) 60

699 1398 (12, 718, 10, 7018) 70

700 1400 (12, 2336) 233

701 1402 (12, 2, 4663) 466

702 1404 (12, 112, 60, 6602) 660

703 1406 (12, 4, 704, 1408) 140

704 1408 (12, 2, 32, 62, 6621) 66

705 1410 (12, 7042) 704

706 1412 (12, 82, 82, 3284) 328

707 1414 (12, 2, 6, 529, 1566) 156

708 1416 (12, 4, 943, 1886) 188

709 1418 (12, 12, 3639) 36

710 1420 (12, 2, 103, 149, 7018) 70

711 1422 (12, 32, 212, 28, 8416) 84

712 1424 (12, 2376) 237

713 1426 (12, 2, 43, 183, 203, 366, 1806) 180

714 1428 (12, 1426) 1426

715 1430 (12, 8417) 84

716 1432 (12, 2, 6, 18, 523, 1562, 4682) 468

717 1434 (12, 1798) 179

718 1436 (12, 32, 4, 122, 2010, 6020) 60

719 1438 (12, 2, 2392, 4782) 478

720 1440 (12, 7192) 719

721 1442 (12, 10, 13011) 130

722 1444 (12, 2, 123, 3639) 36

723 1446 (12, 4, 810, 13610) 136

724 1448 (12, 7232) 723

725 1450 (12, 2, 32, 69, 112, 222, 334, 6618) 66

726 1452 (12, 1450) 1450

727 1454 (12, 1452) 1452

30

728 1456 (12, 2, 43, 4830) 48

729 1458 (12, 56, 232, 11512) 115

730 1460 (12, 4863) 486

731 1462 (12, 2, 2432, 4862) 486

732 1464 (12, 32, 10, 187, 302, 9014) 90

733 1466 (12, 4, 2925) 292

734 1468 (12, 2, 6, 1629) 162

735 1470 (12, 12, 284, 8416) 84

736 1472 (12, 2456) 245

737 1474 (12, 2, 4903) 490

738 1476 (12, 4, 20, 58, 1162, 5802) 580

739 1478 (12, 32, 2107) 210

740 1480 (12, 2, 86, 283, 5624) 56

741 1482 (12, 3704) 370

742 1484 (12, 1482) 1482

743 1486 (12, 2, 43, 6, 103, 122, 18, 206, 302, 362, 604, 902, 1804) 180

744 1488 (12, 7432) 743

745 1490 (12, 7442) 744

746 1492 (12, 2, 32, 62, 352, 702, 1054, 2104) 210

747 1494 (12, 1492) 1492

748 1496 (12, 4, 112, 125, 442, 13210) 132

749 1498 (12, 2, 1669) 166

750 1500 (12, 1498) 1498

751 1502 (12, 18, 392, 2346) 234

752 1504 (12, 2, 6, 832, 1662, 4982) 498

753 1506 (12, 32, 4, 122, 143, 286, 426, 8412) 84

754 1508 (12, 10, 682, 3404) 340

755 1510 (12, 2, 2512, 5022) 502

756 1512 (12, 7552) 755

757 1514 (12, 82, 118, 8816) 88

758 1516 (12, 2, 43, 10015) 100

759 1518 (12, 202, 36, 1808) 180

760 1520 (12, 32, 56, 1512, 212, 10512) 105

31

761 1522 (12, 2, 6, 129, 1569) 156

762 1524 (12, 1522) 1522

763 1526 (12, 4, 20, 6025) 60

764 1528 (12, 2, 5083) 508

765 1530 (12, 10, 138, 6902) 690

766 1532 (12, 1530) 1530

767 1534 (12, 2, 32, 62, 956, 1856) 18

768 1536 (12, 4, 1023, 2046) 204

769 1538 (12, 28, 52, 3644) 364

770 1540 (12, 2, 6, 1828, 5419) 54

771 1542 (12, 112, 6623) 66

772 1544 (12, 7712) 771

773 1546 (12, 2, 43, 512, 1022, 2046) 204

774 1548 (12, 32, 82, 127, 2460) 24

775 1550 (12, 1548) 1548

776 1552 (12, 2, 103, 232, 462, 2306) 230

777 1554 (12, 1948) 194

778 1556 (12, 4, 1552, 6202) 620

779 1558 (12, 2, 6, 1723, 5162) 516

780 1560 (12, 7792) 779

781 1562 (12, 32, 376, 11112) 111

782 1564 (12, 2, 2606) 260

783 1566 (12, 4, 15610) 156

784 1568 (12, 7832) 783

785 1570 (12, 2, 5223) 522

786 1572 (12, 1570) 1570

787 1574 (12, 10, 12, 602, 110, 6602) 660

788 1576 (12, 2, 32, 43, 69, 1220, 203, 6020) 60

789 1578 (12, 18, 82, 7382) 738

790 1580 (12, 5263) 526

791 1582 (12, 2, 56, 86, 106, 4036) 40

792 1584 (12, 7912) 791

793 1586 (12, 4, 3165) 316

32

794 1588 (12, 2, 112, 222, 2532, 5062) 506

795 1590 (12, 32, 226, 6782) 678

796 1592 (12, 143, 36, 2526) 252

797 1594 (12, 2, 6, 18, 583, 1742, 5222) 522

798 1596 (12, 4, 10, 202, 285, 14010) 140

799 1598 (12, 5323) 532

800 1600 (12, 2, 123, 206, 6024) 60

801 1602 (12, 4004) 400

802 1604 (12, 32, 763, 2286) 228

803 1606 (12, 2, 43, 1063, 2126) 212

804 1608 (12, 8032) 803

805 1610 (12, 2018) 201

806 1612 (12, 2, 6, 1783, 5342) 534

807 1614 (12, 5231) 52

808 1616 (12, 4, 810, 18, 362, 7220) 72

809 1618 (12, 2, 32, 62, 103, 212, 306, 422, 2106) 210

810 1620 (12, 1618) 1618

811 1622 (12, 1620) 1620

812 1624 (12, 2, 5403) 540

813 1626 (12, 4, 125, 20, 604, 100, 3004) 300

814 1628 (12, 5423) 542

815 1630 (12, 2, 6, 1809) 180

816 1632 (12, 32, 298, 8716) 87

817 1634 (12, 112, 352, 3854) 385

818 1636 (12, 2, 43, 3645) 36

819 1638 (12, 1636) 1636

820 1640 (12, 10, 148, 7402) 740

821 1642 (12, 2, 5463) 546

822 1644 (12, 56, 52, 2606) 260

823 1646 (12, 32, 4, 122, 232, 694, 922, 2764) 276

824 1648 (12, 2, 6, 18, 609, 1806) 180

825 1650 (12, 82, 4834) 48

826 1652 (12, 718, 12, 8418) 84

33

827 1654 (12, 2, 183, 283, 2526) 252

828 1656 (12, 4, 3011, 6022) 60

829 1658 (12, 9218) 92

830 1660 (12, 2, 32, 62, 3914, 7814) 78

831 1662 (12, 10, 1510, 3050) 30

832 1664 (12, 8312) 831

833 1666 (12, 2, 43, 6, 122, 3645) 36

834 1668 (12, 1666) 1666

835 1670 (12, 1668) 1668

836 1672 (12, 2, 5563) 556

837 1674 (12, 32, 1192, 3574) 357

838 1676 (12, 4, 20, 66, 1322, 6602) 660

839 1678 (12, 2, 123, 149, 8418) 84

840 1680 (12, 98, 112, 9916) 99

841 1682 (12, 202, 8202) 820

842 1684 (12, 2, 6, 86, 103, 244, 302, 4012, 1208) 120

843 1686 (12, 4, 2116, 8416) 84

844 1688 (12, 32, 2470) 24

845 1690 (12, 2, 5623) 562

846 1692 (12, 118, 18, 1988) 198

847 1694 (12, 1692) 1692

848 1696 (12, 2, 43, 2860) 28

849 1698 (12, 8482) 848

850 1700 (12, 5663) 566

851 1702 (12, 2, 32, 69, 187, 547, 1627) 162

852 1704 (12, 12, 130, 7802) 780

853 1706 (12, 4, 56, 1031, 2068) 20

854 1708 (12, 2, 2846) 284

855 1710 (12, 2447) 244

856 1712 (12, 28, 58, 8122) 812

857 1714 (12, 2, 11415) 114

858 1716 (12, 32, 4, 122, 212, 842, 1472, 5882) 588

859 1718 (12, 82, 100, 2008) 200

34

860 1720 (12, 2, 6, 952, 1902, 5702) 570

861 1722 (12, 2158) 215

862 1724 (12, 5743) 574

863 1726 (12, 2, 43, 112, 203, 222, 446, 2206) 220

864 1728 (12, 10, 523, 2606) 260

865 1730 (12, 32, 127, 187, 3642) 36

866 1732 (12, 2, 14412) 144

867 1734 (12, 1732) 1732

868 1736 (12, 4, 346, 6922) 692

869 1738 (12, 2, 6, 9618) 96

870 1740 (12, 232, 36, 8282) 828

871 1742 (12, 1740) 1740

872 1744 (12, 2, 32, 62, 823, 2466) 246

873 1746 (12, 4, 3485) 348

874 1748 (12, 1746) 1746

875 1750 (12, 2, 103, 523, 2606) 260

876 1752 (12, 82, 512, 4084) 408

877 1754 (12, 14612) 146

878 1756 (12, 2, 43, 6, 1247, 18, 3632) 36

879 1758 (12, 32, 505, 15010) 150

880 1760 (12, 8792) 879

881 1762 (12, 2, 5863) 586

882 1764 (12, 143, 202, 14012) 140

883 1766 (12, 4, 8820) 88

884 1768 (12, 2, 56, 106, 183, 9018) 90

885 1770 (12, 28, 60, 4204) 420

886 1772 (12, 32, 10, 112, 302, 334, 1102, 3304) 330

887 1774 (12, 2, 6, 1963, 5882) 588

888 1776 (12, 4, 20, 352, 14012) 140

889 1778 (12, 7424) 74

890 1780 (12, 2, 14812) 148

891 1782 (12, 12, 682, 2048) 204

892 1784 (12, 8912) 891

35

893 1786 (12, 2, 32, 43, 62, 830, 126, 2460) 24

894 1788 (12, 1786) 1786

895 1790 (12, 5963) 596

896 1792 (12, 2, 6, 992, 1988) 198

897 1794 (12, 10, 162, 8102) 810

898 1796 (12, 4, 1792, 7162) 716

899 1798 (12, 2, 2992, 5982) 598

900 1800 (12, 32, 1616, 4832) 48

901 1802 (12, 2572) 25

902 1804 (12, 2, 2524, 5024) 50

903 1806 (12, 4, 18, 362, 342, 6842) 684

904 1808 (12, 12, 138, 2766) 276

905 1810 (12, 2, 6, 18, 669, 1986) 198

906 1812 (12, 3625) 362

907 1814 (12, 32, 212, 367, 2526) 252

908 1816 (12, 2, 43, 103, 206, 1103, 2206) 220

909 1818 (12, 112, 392, 4294) 429

910 1820 (12, 82, 106, 4244) 424

911 1822 (12, 2, 3032, 6062) 606

912 1824 (12, 9112) 911

913 1826 (12, 4, 98, 20, 368, 1808) 180

914 1828 (12, 2, 32, 69, 283, 8420) 84

915 1830 (12, 56, 58, 2906) 290

916 1832 (12, 3056) 305

917 1834 (12, 2, 123, 232, 462, 2766) 276

918 1836 (12, 4, 1832, 7322) 732

919 1838 (12, 10, 832, 8302) 830

920 1840 (12, 2, 6123) 612

921 1842 (12, 32, 1312, 3934) 393

922 1844 (12, 18, 482, 14412) 144

923 1846 (12, 2, 43, 6, 122, 2030, 6020) 60

924 1848 (12, 9232) 923

925 1850 (12, 143, 6023) 602

36

926 1852 (12, 2, 15412) 154

927 1854 (12, 82, 363, 7224) 72

928 1856 (12, 32, 4, 122, 525, 15610) 156

929 1858 (12, 2, 6183) 618

930 1860 (12, 10, 12, 602, 156, 7802) 780

931 1862 (12, 1860) 1860

932 1864 (12, 2, 6, 112, 18, 222, 54, 662, 1982, 5942) 594

933 1866 (12, 4, 3725) 372

934 1868 (12, 1866) 1866

935 1870 (12, 2, 32, 62, 118, 228, 3316, 6616) 66

936 1872 (12, 9352) 935

937 1874 (12, 9362) 936

938 1876 (12, 2, 43, 203, 1003, 5003) 500

939 1878 (12, 1876) 1876

940 1880 (12, 9392) 939

941 1882 (12, 2, 6, 103, 189, 302, 9018) 90

942 1884 (12, 32, 268, 8042) 804

943 1886 (12, 4, 125, 285, 8420) 84

944 1888 (12, 2, 86, 363, 7224) 72

945 1890 (12, 4724) 472

946 1892 (12, 56, 6031) 60

947 1894 (12, 2, 4514, 9014) 90

948 1896 (12, 4, 378, 7562) 756

949 1898 (12, 32, 13514) 135

950 1900 (12, 2, 6, 2109) 210

951 1902 (12, 1900) 1900

952 1904 (12, 10, 172, 8602) 860

953 1906 (12, 2, 43, 718, 1418, 2854) 28

954 1908 (12, 1906) 1906

955 1910 (12, 112, 82, 9022) 902

956 1912 (12, 2, 32, 62, 1221, 212, 422, 8418) 84

957 1914 (12, 2398) 239

958 1916 (12, 4, 1912, 7642) 764

37

959 1918 (12, 2, 6, 18, 352, 702, 2102, 6302) 630

960 1920 (12, 18, 100, 9002) 900

961 1922 (12, 82, 284, 5632) 56

962 1924 (12, 2, 6430) 64

963 1926 (12, 32, 4, 10, 122, 2013, 302, 6026) 60

964 1928 (12, 202, 232, 4604) 460

965 1930 (12, 2, 2149) 214

966 1932 (12, 1930) 1930

967 1934 (12, 6443) 644

968 1936 (12, 2, 43, 6, 122, 149, 2818, 426, 8412) 84

969 1938 (12, 12, 148, 4444) 444

970 1940 (12, 32, 923, 2766) 276

971 1942 (12, 2, 3232, 6462) 646

972 1944 (12, 28, 66, 9242) 924

973 1946 (12, 4, 3885) 388

974 1948 (12, 2, 103, 583, 2906) 290

975 1950 (12, 1948) 1948

976 1952 (12, 9752) 975

977 1954 (12, 2, 32, 56, 69, 106, 1512, 3054) 30

978 1956 (12, 4, 810, 112, 442, 8820) 88

979 1958 (12, 18, 512, 3066) 306

980 1960 (12, 2, 6523) 652

981 1962 (12, 36, 52, 4684) 468

982 1964 (12, 12, 1510, 6030) 60

983 1966 (12, 2, 43, 1303, 2606) 260

984 1968 (12, 32, 704, 2108) 210

985 1970 (12, 10, 178, 8902) 890

986 1972 (12, 2, 6, 98, 18105) 18

987 1974 (12, 1972) 1972

988 1976 (12, 4, 20, 392, 1562, 7802) 780

989 1978 (12, 2, 6583) 658

990 1980 (12, 1978) 1978

991 1982 (12, 32, 943, 2826) 282

38

992 1984 (12, 2, 6603) 660

993 1986 (12, 4, 4445) 44

994 1988 (12, 1986) 1986

995 1990 (12, 2, 6, 86, 129, 2476) 24

996 1992 (12, 10, 18011) 180

997 1994 (12, 9962) 996

998 1996 (12, 2, 32, 43, 62, 126, 1821, 3642) 36

999 1998 (12, 1996) 1996

1000 2000 (12, 3336) 333

References

[1] M. Liebeck, A concise introduction to pure mathematicsThird edition (2010), CRC Press

[2] MATH08004, Group Theory: an Introduction to Abstract MathematicsMain Exam (2008), University of Edinburgh

[3] MATH08004, Group Theory: an Introduction to Abstract MathematicsMain Exam (2011), University of Edinburgh

39