Category Archives: Dice

Dice puzzles (3)


We had two previous posts on dice problems, which you can find here and here.

This is the third post in a small series on dice puzzles. The first one was about the polar bear puzzle and its variations, the second one posed some alternate patterns.
In this post I want to explore some dice puzzles by the British grand master of puzzles, Henry Dudeney.

1) The dice numbers.

I have a set of four dice, not marked with spots in the ordinary way, but with Arabic figures, as shown in the illustration. Each die, of course, bears the numbers 1 to 6. When put together they will form a good many, different numbers. As represented they make the number 1246. Now, if I make all the different four-figure numbers that are possible with these dice (never putting the same figure more than once in any number), what will they all add up to? You are allowed to turn the 6 upside down, so as to represent a 9. I do not ask, or expect, the reader to go to all the labour of writing out the full list of numbers and then adding them up. Life is not long enough for such wasted energy. Can you get at the answer in any other way?
This puzzles was published as puzzle 96 in “Amusement in Mathematics”

You can check your solutions here

2) A trick with dice

The first problem I want to have a look at was published by British puzzler Henry Dudeney in his book “Amusement in Mathematics” (and before that probably in one of the magazines in which he had a monthly column).
I ask you to throw three dice without me seeing them. Then I tell you to multiply the points of the first die by 2 and add 5. Multiply the result by 5 and add the point of the second die. Multiply the result by 10 and add the points on the third die. Mention me the result and I will immediately tell you the points on your three dice.
For example, if you throw 1, 3 and 6, the result will be 386, from which I could at once say what you had thrown. How do I do that?
This puzzles was published as puzzle 386 in “Amusement in Mathematics”

You can check your solutions here

3) The Montenegrin dice game
It is said that the inhabitants of Montenegro have a little dice game that is both ingenious and well worth investigation. The two players first select two different pairs of odd numbers (always higher than 3) and then alternately toss three dice. Whichever first throws the dice so that they add up to one of his selected numbers wins. If they are both successful in two successive throws it is a draw and they try again. For example, one player may select 7 and 15 and the other 5 and 13. Then if the first player throws so that the three dice add up 7 or 15 he wins, unless the second man gets either 5 or 13 on his throw.

The puzzle is to discover which two pairs of numbers should be selected in order to give both players an exactly even chance.

You can check your solutions here

Dice puzzles old and new


Last month we had a look at the most famous of dice puzzles, the Polar bear puzzle. One of the beauties of dice puzzles is that, like playing cards, they are around in many bars.

I found an original dice puzzle in issue 70 of the long defunct British magazine Games & Puzzles, dated May/June 1978.

1) G&P issue 70***/*****



I’m not sure which one is older, the Polar Bears dice puzzle or the one in G&P.

You can check your solution here

2) 4 rolls of 4 dice***/*****



I’m not sure which one is older, the Polar Bears dice puzzle or the one in G&P.

You can check your solution here

Polar bears and other dice puzzles


Polar bears is no doubt the most famous dice puzzle around. I first heard it when I studied mathematics, and Douglas Hofstadters book “Godel, Escher, Bach” may have been the source.
If you want to puzzle your friends, roll 5 dice, and tell the how many polar bears can be spotted. Then roll 5 dice again, let them guess, and tell them the correct number if they guess wrong.

1) Polar bears***/*****
The polar bears puzzle is traditionally presented as a throw of 5 dice. If you are stumped, don’t despair, it is rumored that Bill Gates could only partially solve it.



Even though you may find it hard, I do encourage you to try to solve it before consulting the answer.

You can check your solution here

2) Seals***/*****
Polar bears hunt for seals. How many seals do you count?
This puzzle is inspired by the authors of https://www.pleacher.com/handley/puzzles/polrbear.html.


You can check your solution here

3) Fish***/*****
This puzzle too is inspired by the authors above, though in both instances I changed names to get a more logical picture.


You can check your solution here