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A Very Special Ten-Digit Number

29th June 2008, 04:27 pm

 This puzzle was given to me by John H. Conway, and he heard it from someone else:

Find a ten-digit number with all distinct digits such that the string formed by the first k digits is divisible by k for any k ≤ 10.

Surprisingly, there is a unique solution to this puzzle. Can you find this very special ten-digit number?

For the contrast, consider ten-digit numbers with all distinct digits such that the string formed by the last k digits is divisible by k for any k ≤ 10. These numbers are not so special: there are 202 of them. My puzzle is: find the smallest not-so-special number.

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Category: John Conway, Puzzles, Sequences  |  6 Comments

John Conway’s Wizards

20th June 2008, 03:09 pm

John Conway sent me a puzzle about wizards, which he invented in the sixties. Here it is:

Last night I sat behind two wizards on a bus, and overheard the following:

— A: “I have a positive integral number of children, whose ages are positive integers, the sum of which is the number of this bus, while the product is my own age.”
— B: “How interesting! Perhaps if you told me your age and the number of your children, I could work out their individual ages?”
— A: “No.”
— B: “Aha! AT LAST I know how old you are!”

Now what was the number of the bus?

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Category: John Conway, Math, Puzzles  |  8 Comments

MIT Mystery Hunt Functions

15th February 2008, 05:57 pm

My favorite puzzle at 2008 MIT Mystery Hunt was the puzzle named Functions. Here is this puzzle:

 

36 -> 18      A,B
2 -> 1        A,C,G,H,K,L,O
512 -> 256    A,C,H
4 -> 2        A,G,H,Q
320 -> 160    A,R
411 -> 4      B,E,Q
13 -> 3       B,G,K
88 -> 11      C,D
45 -> 9       C,D,F,J,L
48 -> 6       C,G,M,P,Q
4 -> 1        C,K,L,N,O
36 -> 9       D,E,F
66 -> 8       D,E,G,I
10 -> 3       D,G,L
1 -> 3        D,L
150 -> 15     D,M
3 -> 2        E,H,J,K
25 -> 3       E,K,L,N,Q
9477 -> 14    E,M
129 -> 4      E,N,P
55 -> 10      F,J
411 -> 6      F,K,L,M,N
2002 -> 4     F,O,Q
79 -> 8       G,I,L,P
25 -> 20      H,M
176 -> 80     H,R
3665 -> 8     I,N,Q
7 -> 3        K,Q
11 -> 5       L,M
501 -> 2      L,O,P,Q
8190 -> 5     M,O
180 -> 3      O,P
50 -> 10      R

? -> (?)      F,R
(?) -> ?      J,L
(?) -> ?      A,F
(?) -> ?      N,O,Q
? -> (?)      A,D,J
(?) -> ?      D,H
(?) -> ?      G,K,Q
? -> (?)      B,D,M
(?) -> ?      E,H
? -> (?)      D,F,G,L
? -> (?)      C,G,P
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Category: Math, Puzzle Hunts, Puzzles, Sequences  |  Comment
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