# Term 4 - November 4th (spicy)

##### Maths Share has changed. Each fortnight you will receive a page of problems. There are 3 levels: mild, hot or spicy. Choose which level you’d like to have a go at and place your entry in the NEW entry box at the office! Two winners will be drawn out each week. Happy Maths Sharing!

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

#### Solution

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

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