# Two and Two Poster

## Problem

## Student Solutions

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

**Why?**

F has to be 1 because TWO is less than 1000, so TWO + TWO is less than 2000. This also means that T â‰¥ 5. Note R must be even.

O appears twice, look at the value of O.

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,

$\quad \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{2}$

$\underline{+\, \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{2}}$

$\underline{\, \, \, \text{1}\hspace{1mm} \text{2}\hspace{1mm} \text{U}\hspace{1mm} \text{R}}$

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,

$\quad \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{3}$

$\underline{+\, \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{3}}$

$\underline{\, \, \, \text{1}\hspace{1mm} \text{3}\hspace{1mm} \text{U}\hspace{1mm} \text{R}}$

$\quad \, \, ^1$

then R = 6 and T = 6 which doesn't work.

If O = 4,

$\quad \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{4}$

$\underline{+\, \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{4}}$

$\underline{\, \, \, \text{1}\hspace{1mm} \text{4}\hspace{1mm} \text{U}\hspace{1mm} \text{R}}$

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,

$\quad \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{5}$

$\underline{+\, \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{5}}$

$\underline{\, \, \, \text{1}\hspace{1mm} \text{5}\hspace{1mm} \text{U}\hspace{1mm} \text{R}}$

$\quad \, \, ^1 \, \: ^1$

then R = 0 and T = 7 and we also know that W â‰¥ 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,

$\quad \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{6}$

$\underline{+\, \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{6}}$

$\underline{\, \, \, \text{1}\hspace{1mm} \text{6}\hspace{1mm} \text{U}\hspace{1mm} \text{R}}$

$\quad \, \, ^1$

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,

$\quad \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{7}$

$\underline{+\, \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{7}}$

$\underline{\, \, \, \text{1}\hspace{1mm} \text{7}\hspace{1mm} \text{U}\hspace{1mm} \text{R}}$

$\quad \, \, ^1 \, \: ^1$

then R = 4 and T = 8 and we also know that W â‰¥ 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,

$\quad \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{8}$

$\underline{+\, \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{8}}$

$\underline{\, \, \, \text{1}\hspace{1mm} \text{8}\hspace{1mm} \text{U}\hspace{1mm} \text{R}}$

$\quad \, \, ^1$

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,

$\quad \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{9}$

$\underline{+\, \text{T}\hspace{1mm} \text{W}\hspace{1mm} \text{9}}$

$\underline{\, \, \, \text{1}\hspace{1mm} \text{9}\hspace{1mm} \text{U}\hspace{1mm} \text{R}}$

$\quad \, \, ^1 \, \: ^1$

then R = 8 and T = 9 which doesn't work because O and T are the same.