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
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