# Sunday Times Teaser 2674 – Roll Model

*by Peter G Chamberlain*

I have a board game with squares numbered 1 to 100 on which moves are determined by throwing two dice, one ten sided and numbered 1 to 10, the other four sided and numbered with four primes.

A move consists of throwing the two dice, choosing the the number on either dice or their sum, and then moving forwards or backwards by this number of squares.

After landing on one square, I realised that there were only two squares that could not be reached by any of the possible scores on my next move. Both were prime numbers not on the four sided die.

(a) Which square was I on?

(b) What are the numbers on the four sided die?

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Trying squares around 50 plus and minus we reach the following as a solution:

51+11+/(1-10)=52-72

51-11-/(1-10)=30-50

51+23+/(1-10)=74-84

51-23-/(1-10)=18-28

51+31+/(1-10)=82-92

51-31-/(1-10)=10-20

51+41+/(1-10)=92-100

51-41-/(1-10)=1-10

hence 29 and 73 are the prime numbered squares unreachable with 11,23,31 and 41 being the primes on the four sided die and touching square being 51.

Here’s a solution that entertains the possibility that the unreachable squares could be in the same direction (e.g. if the current square was 52 or greater, and only 2 and 3 were unreachable in backward direction, then all squares in the forward direction would be reachable, so 2 prime squares are unreachable over all). Of course this situation doesn’t happen.

It uses some routines from my

enigma.pylibrary (notably Primes()).Tough puzzle; I got the answer with an hour or two of pen and paper spread over 24 hours, but I wouldn’t care to have to prove uniqueness.

The breakthrough for me – the first one – was realising there are only 54 rolls of the dice available, so there cannot be more than 56 squares in either direction. That immediately gives us that the square is between 44 and 57 (inclusive), which is a start …

I have now confirmed uniqueness by hand, which analysis turned up the bonus information that there is a near-miss to a second solution: the condition that neither missed prime be on the die is required for uniqueness.