Football Sum
Problem
Can you find the values of the nine letters in the sum?
FOOT + BALL GAME
Can you find more than one possibility?
Can you find them all?
Student Solutions
FOOT + BALL = GAME
We had many solutions to this problem worked out using the fact that the letter O has to be zero or nine.
Also if O=0 then $M=L+1$. Therefore $T+L > 11$ implying that $L \leq8$ and $T \geq 3$ . Therefore the only possible $T, L$ pairs are (3,8) (4,7) (4,8) (5,7) (5,8) (6,7) (6,8) (7,4) (7,5) (7,8) (8,3) (8,4) (8,5) (8,6) (9,3) (9,4) (9,5) (9,6) (9,7).
If O=9 then $M=L-1$. Therefore $T+L \leq 8$. We find that the only possible $T, L$ pairs are (1,6) (2,4) (2,5) (3,1) (4,1) (4,3) (5,1) (5,2) (5,3) (6,1) (7,1).
Each member of the class could take on a different T, L pair and you could all pool your solutions.
Systematically checking all these cases, having found O, T, L and M, will easily give all the solutions and we will know there can't be any others. There are 224 different solutions.
A special mention must go to Helen from Maidstone Girls' Grammar School for her very clear explanation of her use of arithmetic and logic to get answers. Michael of Aranmore Catholic Primary School, Western Australia pointed out that there are many solutions to this problem and sent the examples listed below.
1009 2009 1009 7009 1009 5009 1009 6009 2466 1466 7244 1244 5833 1833 6533 1533 ---- ---- ---- ---- ---- ---- ---- ---- 3475 3475 8253 8253 6842 6842 7542 7542 1009 2009 1009 7009 1009 5009 1007 2007 2577 1577 7355 1355 5733 1733 2688 1688 ---- ---- ---- ---- ---- ---- ---- ---- 3586 3586 8364 8364 6742 6742 3695 3695 1009 7009 1007 3007 2990 3990 2009 6009 7466 1466 3688 1688 3155 2155 6144 2144 ---- ---- ---- ---- ---- ---- ---- ---- 8425 8425 4695 4695 6148 6148 8153 8153 1995 4995 1995 4995 1006 4006 8009 1009 4822 1822 4733 1733 4277 1277 1244 8244 ---- ---- ---- ---- ---- ---- ---- ---- 6817 6817 6728 6728 5283 5283 9253 9253 3008 6008 2991 3991 2991 3991 2008 3008 6144 3144 3488 2488 3577 2577 3966 2966 ---- ---- ---- ---- ---- ---- ---- ---- 9152 9152 6479 6479 6568 6568 5974 5974 2004 3004 2003 4003 3911 4991 3995 4995 3677 2677 4788 2788 4066 3066 4211 3211 ---- ---- ---- ---- ---- ---- ---- ---- 5681 5681 6791 6791 8057 8057 8206 8206 etc Alexander of ``Shevah-Mofet'', Tel Aviv also pointed out that there are many solutions and sent us a more interesting problem to which he claims there is only one solution. You might like to try it:
DONALD + GERALD ROBERT