Did you solve it? Give your grey cells a party with the Sweet 16 puzzle

Treat yourself to a cake if you solved this puzzle. Photograph: Food and Drink/Rex

Earlier today I set you this Sweet 16 puzzle:

Did you solve it?

The way to solve this puzzle is to first look at the multiplication and division equations, since there are fewer combinations of numbers that will fit them compared to addition and subtraction.

In the second line there is ODD ÷ ODD = ODD.

By listing the possibilities, this has to be either 15 ÷ 3 = 5 or 15 ÷ 5 = 3. (We can elminate 9 ÷ 3 = 3 since numbers appear only once.)

In the bottom line there is EVEN x EVEN = EVEN, and in the third column ODD x EVEN = EVEN. Since these equations end with the same number (column 3, row 4), we have:

EVEN x EVEN = ODD x EVEN = EVEN

The only possible combination of numbers that will fit this 2 x 6 (or 6 x 2) = 3 x 4 = 12. So now we know that the division must be 15 ÷ 3 = 5, and we can fill in the grid this far:

Half way there

We know that D is either 6 or 2. So A is either 13 or 7. But if A is 7, B is 3, which is forbidden since we have used 3 already. So A is 13, B is 9, C is 6 and D is 2.

The rest of the grid now fills itself, giving the solution as:

Sweet!

I’m not going to solve the the second one completely, but here is how you start:

1. 16 is the highest number. There are only two places it can go, (col 3, row 1) and (col 1 row 2). Placing it anywhere else results in a contradiction.
   
2. Look at (col 3, row 3). It has to be at least 6, since it is the sum of two evens. But we can deduce that it cannot be 12 or 14 (or 16) either.
   
3. And so on, until:
   

Sweet 16 solution Photograph: Michael Polaski

If you want another one to chew on, in this one you don’t know whether the numbers are odd or even. You must still use every number from 1 to 16. I’ll publish the solution in my next column in two weeks.

Click here for a printable version Photograph: Mike Polaski

If you want more puzzles like this, the book Sweet 16 contains 150 of these puzzles and is available in the USA.

I post a puzzle here every second Monday. My most recent book is Snowflake Seashell Star, a colouring book of mathematical images for all ages. (In the US its title is Patterns of the Universe.)

You can check me out on Twitter, Facebook, Google+ and my personal website. And if know of any great puzzles that you would like me to set here, get in touch.