# Sunday Times Teaser 2724 – Headcount

*by Victor Bryant*

A game I play with my grandson has six rounds. In round one, seven coins are tossed, in round two, six, in round three, five, and so on, down to two coins in round six. On each round I then get a number of points equal to the number of heads and my grandson gets a number of points equal to the number of tails. But, before each round, we each predict our score and gain an extra ‘bonus’ of a fixed number of points (less than 40) when our predictions are correct.

In one game we noticed that, at the end of each round, the combined total of our points so far had always been a prime number.

How much is the bonus and what was the sum of our scores at the end of this game?

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Here is a slower but more compact solution:

Here is my variation. As suspected there are many solutions, but only one with a bonus below 40. Interestingly, for any bonus up to 200000, there is only one possible final total.

In fact there is at most 1 solution for any bonus up to 10,000,000

I like your solution Arthur so I have slightly recast it and documented it as follows:

Thanks for making the solution more understandable – I ought to add comments myself.

I made a couple of minor mods to my original version when checking bonuses up to 10,000,000 :

1. Only consider odd values of bonuses, as the total after the third round, 18 + n*bonus must be prime and therefore must be odd (and n must be 1 or 5)

2. An initial check that either 18 + bonus or 18+5*bonus is prime eliminates many other values of bonus

After applying these mods, the time taken by the program was dominated by generating the set of primes.