The 24s Puzzle

Appeared in Volume 8/2, May 1995

Recently my niece Lerisha showed me a mathematical game that they play at school to improve numerical dexterity. You are given four digits in the range 1 to 9. Using each of the four digits once and the operations of addition, subtraction, multiplication, and division (exact integer division only), you have to form an expression that evaluates to 24. Here are some examples copied from the game, with the supplied difficulty rating given from one star (easy) to three star (hard).

1 2 5 6 *	(1+5)*(6-2)
1 2 7 2 *	2*2*(7-1)
2 3 3 8 **	(3-2)*(3*8)
4 6 9 5 **	4+6+(9+5)
7 3 8 3 **	7+8+3*3
7 3 1 6 **	(7-3)*(1*6)
4 8 7 4 ***	4-8+7*4
7 2 4 9 ***	7+9+2*4
2 3 8 9 ***	8//2*(9-3)
3 4 8 1 ***	(3-1)*(4+8)

The children I played with were surprisingly quick at forming an appropriate expression, and some simple heuristics were clearly noticable. I've written a dumb Prolog program to create these expressions, appended below. It's a fun challenge to write a better one that captures some of the obvious heuristics.

go(Numbers):-
  select(N1,Numbers,Numbers1),
  select(N2,Numbers1,Numbers2),
  select(N3,Numbers2,[N4]),
  (0 is N1 mod N2 ->
    member(Op1,[*,//,+,-]);
    member(Op1,[*,+,-])),
  Exp1 =.. [Op1,N1,N2],
  (0 is N3 mod N4 ->
    member(Op2,[*,//,+,-]);
    member(Op2,[*,+,-])),
  Exp2 =.. [Op2,N3,N4],
  (Exp2 > 0 ->
    member(Op3,[*,//,+,-]);
    member(Op3,[*,+,-])),
  Exp3 =.. [Op3,Exp1,Exp2],
  Value is Exp3,
  write(Exp3),write(' '),write(Value),nl,
  Value == 24.

Geoff Sutcliffe
geoff@cs.jcu.edu.au