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THE SPICY PROBLEM: Two and Two
Each of the different letters below stands for a different number.
How many solutions can you find to this cryptarithm?
How can you be sure you have found them all?
Can you create other similar cryptarithms?
Here are some suggestions to start you off.
ONE + ONE = TWO
ONE + TWO = THREE
ONE + THREE = FOUR
FOUR + FIVE = NINE
The first thing we noticed was that F has to be 1 because the most T + T can be is 19 (if you have already carried 1 from the previous column). This also means that T ≥ 5. We also noticed that R must be even.
We decided to look at the value of O again.
If O = 0, then R would also be 0 so that doesn’t work and O can’t be 1 because F = 1.
If O = 2,
TW2
+TW2−−−−−−−
12UR−−−−−−−
then R = 4 and T = 6 and we also know that W < 5 because there can’t be anything carried to the hundreds column. The only possible value of W that hasn’t already been used is 3 but this would mean that U is 6 which is the same as T.
If O = 3,
TW3
+TW3−−−−−−−
13UR−−−−−−−
1
then R = 6 and T = 6 which doesn’t work.
If O = 4,
TW4
+TW4−−−−−−−
14UR−−−−−−−
then R = 8 and T = 7 and we also know that W < 5 because there can’t be anything carried to the hundreds column. So W could be 0, 2 or 3.
W can’t be 0 because then U would be 0 and it can’t be 2 because U would be 4.
If W = 3, U = 6 which works: 734 + 734 = 1468.
If O = 5,
TW5
+TW5−−−−−−−
15UR−−−−−−−
11
then R = 0 and T = 7 and we also know that W ≥ 5 because there has to be 1 carried to the hundreds column.
W can’t be 5 because O = 5.
If W = 6, U = 3 which works: 765 + 765 = 1530.
If W = 7, U = 5 which doesn’t work because O and U are the same.
If W = 8, U = 7 which doesn’t work because T and U are the same.
If W = 9, U = 9 which doesn’t work because W and U are the same.
If O = 6,
TW6
+TW6−−−−−−−
16UR−−−−−−−
1
then R = 2 and T = 8 and we also know that W < 5 because there can’t be anything carried to the hundreds column. So W could be 0, 3 or 4.
If W = 0, U = 1 which doesn’t work because F and U are the same.
If W = 3, U = 7 which works. 836 + 836 = 1672
If W = 4, U = 9 which works. 846 + 846 = 1692
If O = 7,
TW7
+TW7−−−−−−−
17UR−−−−−−−
11
then R = 4 and T = 8 and we also know that W ≥ 5 because there has to be 1 carried to the hundreds column.
If W = 5, U = 1 which doesn’t work because F and U are the same.
If W = 6, U = 3 which works. 867 + 867 = 1734
W can’t be 7 because O = 7.
If W = 8 , U = 7 which doesn’t work because O and U are the same.
If W = 9, U = 9 which doesn’t work because W and U are the same.
If O = 8,
TW8
+TW8−−−−−−−
18UR−−−−−−−
1
then R = 6 and T = 9 and we also know that W < 5 because there can’t be anything carried to the hundreds column. So W could be 0, 2, 3 or 4.
If W = 0, U = 1 which doesn’t work because F and U are the same.
If W = 2, U = 5 which works: 928 + 928 = 1856.
If W = 3, U = 7 which works: 938 + 938 = 1876.
If W = 4, U = 9 which doesn’t work because T and U are the same.
If O = 9,
TW9
+TW9−−−−−−−
19UR−−−−−−−
11
then R = 8 and T = 9 which doesn’t work because O and T are the same.
So there are seven possible answers:
938+938=1876
928+928=1856
867+867=1734
846+846=1692
836+836=1672
765+765=1530
734+734=1468