Do you remember how to divide three apples among four people? Make apple sauce, of course. In the following two puzzles you are not allowed to cut apples. Here is an old riddle:

There are four apples in a basket. How can you divide them among four people, so that one apple remains in the basket?

Here is a variation from Konstantin Knop’s blog:

There are four apples in a basket. How can you divide them among three people, so that no one has more than the others and one apple remains in the basket?

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1. #### CalcDave:

Put a person in the basket?

2. #### Gregory Marton:

For the first one, perhaps we can go bite by bite? For the second one, I don’t get how it’s not obvious: each gets one apple, and one remains in the basket. Help?

3. #### Tanya Khovanova:

Gregory,

There is more to the second one.

4. #### Asher:

For the first one, the most straightforward solution is to give three people apples, and the fourth person the basket with one apple in it.

For the second one, the intuitive response is Gregory’s answer – each person takes one apple and leaves one in the basket.

In the scenario in which the three people are good friends, even though they’re holding different numbers of apples, it could be said that they each have the same number of apples. Trivially, they hold three and leave the last in the basket.

In the scenario in which the three people are enemies, they will put one apple in the basket. Then they will fight over how to divide the other apples and the apples will all be discarded in the process.

The two above scenarios are clearly exhaustive, and hence the apples will be evenly divided.

Alternatively, you could leave all four apples in the basket. It can be said that one apple remains in the basket – one apple, and one apple, and one apple, and one apple. The problem does not specify that only one apple must be left behind.

5. #### Math Fan:

These sound like tricky wording rather than math problems.

For the first puzzle, give out 1 apple, 1 apple, 1 apple and 1 basket with an apple. They are equally divided and one is still in the basket.
For the second puzzle, give out 2 apples, 1 apple and 1 basket with an apple. The first person has 2 apples which is not more than the others (because between the others they have 2 apples). The second has 1 apple which is not more than the others. The last has 1 apple and basket which again is not more than the others. Again there is one apple left in the basket.

6. #### ed:

Since we’re considering tricky wording. The requirement that apples should be *equally* divided is not made explicit.

7. #### jhanson:

off the topic, i have been looking for a good explanation of why a negative times a negative equals a positive. neg 5 dollars in my checking account times negative 10 dollars in your checking account certainly does not equal positive 50 dollars for us to split. I dont understand how nothing x nothing equals something?? can anyone help me with this?

8. #### Animesh:

had come across the same puzzle with apples replaced with eggs few months back…but anyway approach is same as @MathFan pointed out…

9. #### Joseph:

jhanson: If you put -10 dollars into your account each day, then -5 days from now, you’ll have 50 more dollars in your account than you do now. (Or, more precisely: if you take 10 dollars out of your account each day, then 5 days ago, you had 50 more dollars in your account than you do now.)

10. #### Paul:

Person C eats 3 apples, leaving one in the basket.
Person B eats Person C
Person A eats Person B

11. #### Animesh:

@Paul now we know u are a cannibal ðŸ˜‰