MONEY
−SEND
----------
MORE
is equivalent to
MORE
+SEND
----------
MONEY
Right off the bat, M has to be 1, because the sum of any two single-digit numbers is at most 18.
1ORE
+SEND
----------
1ONEY
Now, since the last column is 1 + something...and the 1 + something must be a 2-digit number... S must be 9 and O must be 0.
10RE
+9END
----------
10NEY
Now since we have 0, E, and N in the 2nd column, we know that there must be a carry from the previous column (otherwise the two letters would have to be the same). So 1 + E = N.
But then looking at third column, we see that either R + N = E (or R + 1 + N = E). Since E is one less than N, either R = 9 (no carry) or R = 8 (carry).
And then, honestly, I think this is as far as you can keep reasoning.
Which means, I think that, there are lots of answers.
10RE
+9END
----------
10NEY
If R = 8, then E + D ≥ 10 (because we need the carry). Since N also must be 1 more than E, we could have 36 solutions!
M O N E Y S D R
1 0 2 1 0 9 9 8: (10,210 - 9,129 = 1,081)
1 0 3 2 1 9 9 8: (10,321 - 9,239 = 1,082)
1 0 3 2 0 9 8 8: (10,320 - 9,238 = 1,082)
1 0 4 3 2 9 9 8: (10,432 - 9,349 = 1,083)
1 0 4 3 1 9 8 8: (10,431 - 9,348 = 1,083)
1 0 4 3 0 9 7 8: (10,430 - 9,347 = 1,083)
1 0 5 4 3 9 9 8: (10,543 - 9,459 = 1,084)
1 0 5 4 2 9 8 8: (10,542 - 9,458 = 1,084)
1 0 5 4 1 9 7 8: (10,541 - 9,457 = 1,084)
1 0 5 4 0 9 6 8: (10,540 - 9,456 = 1,084)
1 0 6 5 4 9 9 8: (10,654 - 9,569 = 1,085)
1 0 6 5 3 9 8 8: (10,653 - 9,568 = 1,085)
1 0 6 5 2 9 7 8: (10,652 - 9,567 = 1,085)
1 0 6 5 1 9 6 8: (10,651 - 9,566 = 1,085)
1 0 6 5 0 9 5 8: (10,650 - 9,565 = 1,085)
1 0 7 6 5 9 9 8: (10,765 - 9,769 = 1,086)
1 0 7 6 4 9 8 8: (10,764 - 9,768 = 1,086)
1 0 7 6 3 9 7 8: (10,763 - 9,767 = 1,086)
1 0 7 6 2 9 6 8: (10,762 - 9,766 = 1,086)
1 0 7 6 1 9 5 8: (10,761 - 9,765 = 1,086)
1 0 7 6 0 9 4 8: (10,760 - 9,764 = 1,086)
1 0 8 7 6 9 9 8: (10,876 - 9,789 = 1,087)
1 0 8 7 5 9 8 8: (10,875 - 9,788 = 1,087)
1 0 8 7 4 9 7 8: (10,874 - 9,787 = 1,087)
1 0 8 7 3 9 6 8: (10,873 - 9,786 = 1,087)
1 0 8 7 2 9 5 8: (10,872 - 9,785 = 1,087)
1 0 8 7 1 9 4 8: (10,871 - 9,784 = 1,087)
1 0 8 7 0 9 3 8: (10,870 - 9,783 = 1,087)
1 0 9 8 7 9 9 8: (10,987 - 9,899 = 1,088)
1 0 9 8 6 9 8 8: (10,986 - 9,898 = 1,088)
1 0 9 8 5 9 7 8: (10,985 - 9,897 = 1,088)
1 0 9 8 4 9 6 8: (10,984 - 9,896 = 1,088)
1 0 9 8 3 9 5 8: (10,983 - 9,895 = 1,088)
1 0 9 8 2 9 4 8: (10,982 - 9,894 = 1,088)
1 0 9 8 1 9 3 8: (10,981 - 9,893 = 1,088)
1 0 9 8 0 9 2 8: (10,980 - 9,892 = 1,088)
If R = 9, then Y = E + D ≤ 9 (no carry):
M O N E Y S D R
1 0 1 0 0 9 0 9: (10,100 - 9,010 = 1,090)
1 0 1 0 1 9 1 9: (10,101 - 9,011 = 1,090)
1 0 1 0 2 9 2 9: (10,102 - 9,012 = 1,090)
1 0 1 0 3 9 3 9: (10,103 - 9,013 = 1,090)
1 0 1 0 4 9 4 9: (10,104 - 9,014 = 1,090)
1 0 1 0 5 9 5 9: (10,105 - 9,015 = 1,090)
1 0 1 0 6 9 6 9: (10,106 - 9,016 = 1,090)
1 0 1 0 7 9 7 9: (10,107 - 9,017 = 1,090)
1 0 1 0 8 9 8 9: (10,108 - 9,018 = 1,090)
1 0 1 0 9 9 9 9: (10,109 - 9,019 = 1,090)
There are 9 solutions for N = 1, E = 0, 8 solutions for N = 2, E = 1, 7 for N = 3, E = 2...
9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 more solutions.
Altogether, I think this problem has 36 + 45 = 81 possible answers.
Even if you specify that each number must be different there are many solutions.
2007-01-29 15:22:25
·
answer #1
·
answered by Jim Burnell 6
·
1⤊
0⤋