## Teaser 2720: Better tetra

**From The Sunday Times, 9th November 2014** [link] [link]

I have four tetrahedral dice. Each has four identical triangular faces and on each face of each die is one of the numbers 1, 2, 3 or 4 (repeats being allowed on a die). No die uses the same four numbers and, for each die, the sum of its four numbers is ten.

I play a game with my friend. My friend chooses a die first and then I choose one of the remaining three dice. We each throw our chosen die and score the face-down number: sometimes it’s a draw, otherwise the higher number wins. I can choose my die so that I am always more likely to win.

So my friend changes the rules. We now throw our chosen die twice, add the two numbers, and the higher total wins.

Now he knows that there is one die he can choose that makes him more likely to win the game.

What are the four numbers on the winning dice?

The wording for this puzzle has been modified from the originally published version.

[teaser2720]

## Jim Randell 10:00 am

on30 June 2022 Permalink |(See: [@youtube]).

We assume that the dice are chosen in such a way that each player is aware of which die has been chosen. (So, for example, they may be different colours and arranged on the table, but not be hidden a bag and chosen by putting your hand in and selecting one at random).

When we compare 2 dice, we say that one is “better” than the other, if it is more like to win a game against the other die.

There are 5 possible dice (i.e. 5 possible decompositions of 10 into the numbers 1, 2, 3, 4), but the set only contains 4 of these.

It turns out there is only one possible set of 4, where whatever die A chooses B can always choose a better die.

And from this set we can look to see if there is die which is better when played against each of the other dice.

This Python program runs in 61ms. (Internal runtime is 593µs).

Run:[ @replit ]Solution:The best die in the second game is: (1, 3, 3, 3).The four dice are:

And we have:

So whichever die is chosen first, there is always a choice from the other three that is better.

LikeLike

## Frits 5:54 pm

on30 June 2022 Permalink |Apparently we can also compare lists with the less than operator in Python.

LikeLike