There are no solutions because it can’t be done! No matter what combination you try, adding 10 of these odd numbers together will never give you 41. Did you notice anything about the answers you were getting? The answer should always have been an EVEN number. (An even number is one that can be exactly divided by 2).

If you add an EVEN number of ODD numbers together you will always get an EVEN answer. You can try this out for yourself.

Why does this happen? It’s easier to explain if we show numbers as blocks.

Take the number 7. We can draw this as 7 blocks.

Let us now pair the blocks up in twos.

You can see that we have one odd (unpaired) block and that this will be true for every ODD number that we draw.

If we now add TWO odd numbers together, for example 7 and 3

You can see that the unpaired block from each odd number is now paired and that the number produced is an even number (10 in this example).

This will happen every time you add together an even number of ODD numbers. Each unpaired block will be able to pair with one other, leaving no unpaired blocks, so producing an EVEN number. If however you add together an odd and an even, an ODD number is always produced because there is no unpaired block on the even number to pair with the left-over one from the odd number.

So the rules are:

ODD + ODD = EVEN

ODD + EVEN = ODD

EVEN + EVEN = EVEN

And an

Odd number of ODDS added together = ODD
Even number of ODDS added together = EVEN
Even number of EVENS added together = EVEN
Odd number of EVENS added together = EVEN

Now try this:

would you get an odd or an even number answer from:

ODD + ODD + ODD =

ODD + EVEN + ODD =

EVEN + EVEN + EVEN + ODD + ODD =

ODD + EVEN + ODD + EVEN + ODD =

This puzzle has been adapted from the Nrich Workshop. See the nrich website at www.nrich.maths.org for more activities, puzzles and mathematical resources.