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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