Dear Uncle Colin,

My teacher recently challenged me to tackle the missing numbers problem below and I don’t know where to start! It’s driving me to disgust with the whole number system.

Crazy Old Numbers, Wacky And Yucky

Dear CONWAY,

That feeling of disgust is a completely normal part of the problem-solving process – it goes away once you figure out the answer!

Now, as for getting started, my approach is to replace all of the boxes with letters, like so: . ABC . ×D5 ===== 22EF GHIJ ===== 2011K

What this is showing is a ‘traditional’ column-method multiplication sum: above the first line are the two numbers you’re multiplying together; immediately below that are the top number multiplied by the second digit (5) of the second row, followed by the top number multiplied by the first digit (D) of the second row, moved one space to the left. These two are then added up below the second horizontal line to get the answer, 2011K.

There are a few things that jump out at me. Firstly, F and K are clearly the same – and (because of the 5) must be 0 or 5. Secondly, G has to be either 1 or 2, H has to be 7 or 8, and I has to be 8 or 9, just to make the bottom two lines add up. Whichever of the two I is, there has to be a carry, so H is 7; that means there’s also a carry into the ‘G’ column, so G is 1. We’ve now got:

. ABC . ×D5 ===== 22EF 17IJ ===== 2011K

Now for the clever bit. We know that ABC × 5 is between 2200 and 2299, so 440 ≤ ABC < 460. A is definitely 4, and B is either 4 or 5. What do you multiply 440-something or 450-something to get 1700-and-something? It has to be 4 – so D is clearly 4.

At this point, the puzzle is pretty much done! We know the number in the second row is 45, which is a multiple of 9, so 2011K has to be a multiple of 9 and therefore the sum of its digits is 9 (or a multiple thereof). In particular, K has to be 5, as does F.

We know that 4BC × 45 = 20115, so 4BC = 20115 ÷ 45. You might try long division there; the Mathematical Ninja would clear his throat and remind you to double them both to make the numbers better: 40230 ÷ 90 = 4023 ÷ 9. That works out to be 447.

The rest is filled in automatically:

. 447 . ×45 ===== 2235 1788 ===== 20115

Still disgusted? I think that’s pretty nice.

-- Uncle Colin