T | H | R | E | E | |

T | H | R | E | E | |

+ | F | O | U | R | |

E | L | E | V | E | N |

Can you find digits to replace the letters to make this sum correct?

Note to teacher: If you are presenting this puzzle to your class using a projector you can slowly scroll down the step by step guide below giving clues that might enable your students to finish off the puzzle for themselves.

There is no standard way to solve a problem like this. Each one is quite different in structure and will have its own clues; but having said that here is an example of how to solve this particular challenge which may give you ideas for solving others.

We should assume that each letter stands for a different digit.

1. The first thing to notice is that the bottom line has six digits. When the two Ts are added together the answer caused a digit to be carried. This carried digit is most likely a one though there is a small possibility it could be a 2. Lets go with one and backtrack to here if it does not work out.

T | H | R | 1 | 1 | |

T | H | R | 1 | 1 | |

+ | F | O | U | R | |

1 | L | 1 | V | 1 | N |

2. Looking at the tens column, to produce a digit one in the bottom line, the U must be a nine unless there is a digit carried from the units column. Lets go with nine and backtrack to here if it does not work out.

T | H | R | 1 | 1 | |

T | H | R | 1 | 1 | |

+ | F | O | 9 | R | |

1 | L | 1 | V | 1 | N |

It may be useful to keep a record of the letters involved in the puzzle and the possible digits they could stand for:

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

T | ✗ | ✗ | ||||||||

H | ✗ | ✗ | ||||||||

R | ✗ | ✗ | ||||||||

E | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

F | ✗ | ✗ | ||||||||

O | ✗ | ✗ | ||||||||

U | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ |

L | ✗ | ✗ | ||||||||

V | ✗ | ✗ | ||||||||

N | ✗ | ✗ |

3. Now it is a little more difficult to spot any certainties though we can see that if R is even then N is also even (and the inverse).

Let's make a guess and see how far it takes us. We'll start with the units column as there won't be any carried numbers to worry about.

Our guess for R will be the smallest digit it could possibly be, zero. If that is not correct and we get stuck further along we can always backtrack to this point and change our guess to the next smallest digit and so on.

If R is zero then N must be 2.

At this point be careful you don't get the letter O and the digit zero confused. We will use the slashed zero, Ø, to avoid confusion.

T | H | Ø | 1 | 1 | |

T | H | Ø | 1 | 1 | |

+ | F | O | 9 | Ø | |

1 | L | 1 | V | 1 | 2 |

The table is updated with this information.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

T | ✗ | ✗ | ✗ | ✗ | ||||||

H | ✗ | ✗ | ✗ | ✗ | ||||||

R | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

E | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

F | ✗ | ✗ | ✗ | ✗ | ||||||

O | ✗ | ✗ | ✗ | ✗ | ||||||

U | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ |

L | ✗ | ✗ | ✗ | ✗ | ||||||

V | ✗ | ✗ | ✗ | ✗ | ||||||

N | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

By working through the columns from right to left we can deal with the carried numbers as they appear.

4. Let's make another guess. This time for the letter O. We can see from the table that the smallest digit it could possibly be is a three. If that is not correct and we get stuck further along we can always backtrack to this point and change our guess to the next smallest digit and so on.

If O is 3 then V must be four as one is carried over from the 10s column.

T | H | Ø | 1 | 1 | |

T | H | Ø | 1 | 1 | |

+ | F | 3 | 9 | Ø | |

1 | L | 1 | 4 | 1 | 2 |

The table is updated with this information.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

T | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||||

H | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||||

R | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

E | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

F | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||||

O | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

U | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ |

L | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||||

V | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ |

N | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

5. Let's make another guess. This time for the letter H. We can see from the table that the smallest digit it could possibly be is a five.

If H is five, F would have to be one to produce the given column total but we have already assigned the digit one so H cannot be five. Let's backtrack.

We can see from the table that the next smallest digit it could possibly be is a six.

If H is six, F would have to be nine to produce the given column total but we have already assigned the digit nine so H cannot be six. Let's backtrack.

We can see from the table that the next smallest digit it could possibly be is a seven.

If H is seven, F would have to also be seven to produce the given column total but we cannot assign the same digit to two different letters. Let's backtrack.

We can see from the table that the only other possible choice for the letter H is eight.

If H is eight, F would have to be five:

T | 8 | Ø | 1 | 1 | |

T | 8 | Ø | 1 | 1 | |

+ | 5 | 3 | 9 | Ø | |

1 | L | 1 | 4 | 1 | 2 |

The table is updated with this information.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |

T | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||

H | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ |

R | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

E | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

F | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ |

O | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

U | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✓ |

L | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ||

V | ✗ | ✗ | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ |

N | ✗ | ✗ | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ |

6. Let's now guess the letter T. We can see from the table that the smallest digit it could possibly be is a six.

Two sixes with the two carried from the thousands column makes 14 but L cannot be four as that digit has already been assigned. Let's backtrack.

The only remaining possibility is that T is seven making L equal to six.

7 | 8 | Ø | 1 | 1 | |

7 | 8 | Ø | 1 | 1 | |

+ | 5 | 3 | 9 | Ø | |

1 | 6 | 1 | 4 | 1 | 2 |

Here are some other solutions found by making different guesses in the process above. Can you find any others?

8 | 4 | Ø | 1 | 1 | |

8 | 4 | Ø | 1 | 1 | |

+ | 3 | 5 | 9 | Ø | |

1 | 7 | 1 | 6 | 1 | 2 |

7 | 4 | 6 | 1 | 1 | |

7 | 4 | 6 | 1 | 1 | |

+ | 2 | Ø | 9 | 6 | |

1 | 5 | 1 | 3 | 1 | 8 |

4 | 6 | 5 | 1 | 1 | |

4 | 6 | 5 | 1 | 1 | |

+ | 8 | 2 | 9 | 5 | |

1 | Ø | 1 | 3 | 1 | 7 |

Here are some other word sums to try:

Finally here is the URL which will take you to a code cracking activity.

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