Neurobics

The Soma Cube

by Russell Dear

The Soma cube is well known to many people, recreational mathematicians in particular. It is a puzzle designed by Piet Hein consisting of seven pieces made from cubes. Six of the pieces are formed from four cubes and one from three, as shown.

The main purpose of the puzzle is to arrange the seven pieces to form a 3 by 3 by 3 cube.

A new problem associated with the Soma cube is to find all possible parallelopipeds (rectangular solids) using some or all of the seven pieces. Obviously a 3 by 3 by 3 cube can be made but how many different rectangular solids can one make under the given conditions?

Strange Dice

Late one evening after a hard day's tramping my companion asked me if I'd like a game of dice. "Sure", I said, "let's play craps." My friend fumbled about in her pack and finally produced two unusual dice. One had its six faces numbered 1, 2, 2, 3, 3, 4, the other 1, 3, 4, 5, 6, 8. Craps is scored by adding the numbers which appear uppermost when two dice are thrown and depends upon the various possible scores occurring in certain proportions. I hesitated for a minute or two and then said, "OK then, let's play craps."

It's obvious why I hesitated, but why did I eventually agree to play?

Goldbach's Conjecture

Number theory is a fascinating branch of mathematics full of delightful outcomes, many unproven, and a rich field for obscure research. One of it's results is called Goldbach's conjecture which states that all even numbers, except 2, can be expressed as the sum of two primes. Prime numbers have only two divisors (1 is not prime but 2, 3, 5, 7, and 11 are). For example 6 = 3 + 3, 8 = 3 + 5, and so on. Often an even number can be expressed as the sum of two primes in more than one way called it's Goldbach count.

10 has a Goldbach count of 2 since 10 = 3 + 7 = 5 + 5, while 30 = 7 + 23 = 11 +19 = 13 + 17 has a Goldbach count of 3. Here are two problems for you to consider:

(a) 30 has a Goldbach count of 3 while 60 has one of 6 and 90 one of 9. Does the pattern continue? Can you prove it?

(b) What (even) number less than 100 has the highest Goldbach count?

Answers

1) Six -- 1x3x5, 2x2x3, 2x2x4, 2x2x5, and 2x3x4 as well as 3x3x3.
2) Because the unusual dice actually give the same set of possible scores as conventional dice, and in the correct proportions!
3a) 120 certainly has a Goldbach count of 12 but there the pattern ends. It is sheer coincidence -- the Goldbach count of 150 is also 12.
(3b) 90.

Russell Dear is a Mathematician living in Invercargill