Modular arithmetic can work, but you need to be a little bit careful. If you want a number from 0-10 then modulo 11 works perfectly (as long as you pick a numb

Modular arithmetic can work, but you need to be a little bit careful. If you want a number from 0-10 then modulo 11 works perfectly (as long as you pick a number well outside any 11 times table you have memorised). This is because it's relatively easy to pick a number without immediately knowing what the mod 11 version of it will be. For 1-10 you could do mod 10, but that's too easy to cheat. Instead you can take you initial guess n, and multiply (n mod 5) + 1 by (n mod 2) + 1. For example 42 -> ((42 mod 5) + 1) x ((42 mod 2) + 1) = 3 x 1 = 3; 43 -> 4 x 2 = 8. This second scheme works because we're using a factorisation of the length of our intial range of guesses, and then multiplying the answers. We add 1 because 0s make multiplying boring (and we want 1-10 in any case). There will still be bias, especially if you've used the scheme before. With the mod5 mod2 scheme the biggest bias is that you only get a 1 if the original guess is congruent to 0 mod 10. A way to avoid that bias, which unfortunately introduces another bias, is to do a two modulo steps. Start with a biggish number, and then do your final one. This is harder to do in your head and makes some final guesses more likely, but only a bit. So for example, do final guess = (n mod 19) mod 10 + 1. 19 is easyish to calculated because you find the closest multiple of 20, workout that mod 19, and add any leftovers to it. 42 mod 19 = 40 mod 19 + 2 = 2 + 2 = 4 (42 mod 19) mod 10 + 1 = 4 mod 10 + 1 = 5