## Fractional puppies

By David Shaw, 27 June 2018

Kelly has a pet store. She recently got in a whole litter of puppies and tried to sell them all.

- On the first day, she sold half of the puppies, plus half a puppy.
- On the second day, she sold one third of the remaining puppies, plus one third of a puppy.
- On the third day, she sold half the puppies she had left, plus half a puppy.
- Then she only had one puppy left!

How many puppies were in the litter? (Hint: Kelly didn’t sell any fractions of puppies!)

Scroll down for the answer!

## Brainteaser answer

How can Kelly sell half or a third of a puppy? She can’t, and that’s the key to understanding this problem.

Kelly can only sell exactly half of her puppies if she has an even number of puppies. If she has, say five puppies, then half of five is two and a half, which is not a whole number. If she actually sold three puppies, that would be half of her puppies (2.5), plus another half a puppy.

With this in mind, we can tackle the problem. It’s best done backwards.

### Third day

On the third day, Kelly sold half her puppies plus a half, and was left with one puppy. She must have started the day with an odd number of puppies. We also know that one is a bit less than half the starting number. A guess and check shows that she started the third day with three puppies.

### Second day

The second day is trickier, but similar. This time, rather than looking for odd numbers of puppies, look for numbers that aren’t divisible by three. A bit of trial and error will find:

1/3 of 5 = 1 2/3

1 2/3 + 1/3 = 2 puppies sold

5 – 2 = 3 puppies left for day three

So she started day two with five puppies.

### First day

Finally, day one is relatively simple. The number of puppies must be odd, and slightly more than double five. If we try 11 puppies we find that she sold six, and ended up with five. This works with the other numbers.

### Final answer

To sum up:

- Kelly started with 11 puppies.
- She sold six on the first day, leaving five.
- She sold two on the second day, leaving three.
- She sold two on the third day, leaving one.

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