# Ghost Equation Quiz Puzzle - Answer

This week's puzzle was two equations that had to be solved. Unlike some of the previous puzzles that look simple but turn out to be hard, this one looked impossible until you narrowed down the possible options and then it became easier - not easy, just easier.

The two equations are:

□ x □ = □□

□ x □ = □□

Each of the □ symbols represents one of the digits 1 - 9 and none of the digits is repeated. So use each digit from 1 to 9 once and once only!

And, of course, you have to make both equations correct.

As we hinted, you need to narrow down the options, and the most productive approach is to work out where the 5 could go.

It can't be any of the three single-digit numbers, as multiplying 5 by an even number will result in a number ending in 0 (and there is no zero), and multiplying by an odd number will result in a number ending in 5 (and that would need two 5s).

By the same logic it can't be the least significant digit of the two-digit multiplier either.

And it can't be the most significant digit of the two-digit multiplier, as multiplying by 2 or above would give a three-digit product. And clearly you can't multiply it by 1.

Lastly, the least significant digits of the two products can't be 5, because one of the two multipliers to produce it would also have to be, or end in, 5, requiring two 5s.

So, from all the above, the 5 must be the most significant digit of one of the products. That is, one of the products must be 50-something.

The options are:

3 x 17 = 51 (not allowed because of the two 1s)

4 x 13 = 52 (leaving 6, 7, 8 and 9)

3 x 18 = 54 (leaving 2, 6, 7 and 9)

4 x 14 = 56 (not allowed because of the two 4s)

3 x 19 = 57 (leaving 2, 4, 6 and 8)

2 x 29 = 58 (no allowed because of the two 2s)

And

6 x 9 = 54 (leaving 1, 2, 3, 7 and 8)

7 x 8 = 56 (leaving 1, 2, 3, 4 and 9)

After a bit of trial and error on these five options you will arrive at

6 x 9 = 54

3 x 27 = 81

which is the only solution.