# Math Is Fun Forum

Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

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## #1 2021-11-08 06:00:01

T897T
Member
Registered: 2021-09-21
Posts: 42

### 21 point die puzzle

I have been wondering about this for some time but I will not post this in the help me area so those there don't think I need help.
The sides of a typical 6 sided die add up to 21. So with a 6 sided die whose sides add up to 21, what is the highest probability you can get to win in a die rolling contest against a normal die. Ex 0,0,3,6,6,6 has a 17/36 chance of winning. You can not put negative numbers, and I see no point of using numbers above 7.

Forced purity isn't pure.
Good and Evil are different for each person.
I chose 8 9 and 7 randomly.

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## #2 2021-11-08 22:21:31

Bob
Registered: 2010-06-20
Posts: 9,649

### Re: 21 point die puzzle

hi T897T

I see no point of using numbers above 7

I'm wondering if you meant above 6 here?  If your modified die is {7,7,7,7,7,7} then you win every time.  If {6,6,6,6,6,6} then the probability is 30/36.

Am I following your post correctly?

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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## #3 2021-11-09 08:17:17

T897T
Member
Registered: 2021-09-21
Posts: 42

### Re: 21 point die puzzle

The sides can be above 7 but the total must be 21 and no side may be negative.
The reason I said there is no point is because 8 and 7 give you the same chance of winning so does 9 and so on.
Sides can also be 0

Forced purity isn't pure.
Good and Evil are different for each person.
I chose 8 9 and 7 randomly.

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## #4 2021-11-09 21:37:23

Bob
Registered: 2010-06-20
Posts: 9,649

### Re: 21 point die puzzle

Arggh, the penny drops.  I hadn't realised you wanted a 21 limit.  So far {000777} seems best at P = 0.5  I'm working on a proof.

LATER EDIT:

If you have {000777} then 7 beats everything and zero beats nothing so the probability of a win is 18/36

I want to show that if you change a 7 to other numbers that add up to 7 the number of wins goes down.

General theorem.  A number n beats everything lower so the number of wins associated with n is (n-1)

If you change n to a lower number m:

There are two  new numbers generated by this: n-m and m

The win total from n-m is (n-m-1) and from m is (m-1) so the new total is (n-m-1) + (m-1) = n - 2 so the change has caused the total wins to go down (n-1 drops to n - 2)

This happens whenever you substitute a lower pair of numbers. So reducing a 7 will lower the win total.  So {000777} is best.

example {002577} The win total is now 1 + 4 + 6 + 6 = 17.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

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