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Got this question at work - seems simple enough but I cant work it out. Please help:

It was Easter Sunday and three friends went on an Easter egg hunt. Between them Jo, Harry and Greg collected a pile of Easter Eggs.

After all their hunting they were very tired and went straight to bed.

During the night Jo woke and took one third of the eggs, but she had to eat one before she could take exactly one third. Jo went back to bed.

Later Harry woke and took one third of the eggs that were left but he had to eat one before he could take exactly one third. Harry went back to bed.

Then much later Greg woke and took one third of the eggs that were left but he had to eat one before he could take exactly one third. Greg went back to bed.

For breakfast in the morning, the children were able to equally share all the eggs that were left in the original pile.

  1. How many eggs might they have originally found?

  2. How many eggs did each child get to eat for breakfast?

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We can work backwards to find a solution that satisfies all the conditions.

At breakfast, three people split the eggs evenly, so the number of eggs must have been divisible by 3 at this point. In addition, before breakfast, exactly $\frac{1}{3}$ were removed, so the number of eggs at breakfast must have been divisible by 2, because:

$$ \frac{2}{3} \text{eggs before breakfast} = \text{eggs at breakfast}$$

$$\text{eggs before breakfast} = \frac{3}{2} \text{eggs at breakfast}$$

For example, if there were 5 eggs at breakfast (not divisible by 2), there must have been 7.5 eggs before breakfast, since removing one third of 7.5 yields 5 eggs. However, it's impossible to have half an egg, which means there could not have been 5 eggs at breakfast.

Now, we know that the eggs at breakfast must have been divisible by 2 and by 3. The lowest number that satisfies these conditions is 6.

That means Greg must have had 10 eggs, to eat 1 and take one third of 9 to yield 6.

Harry, therefore, must have had 16 eggs, to eat 1 and take one third to yield 10.

Lastly, Jo must have had 25 eggs, to eat 1 and take one third to yield 16.

Note that 10, 16, and 25 are all indivisible by 3, which is a sensible check because the story says no one could take one third of the eggs before eating 1. Note that 9, 15, and 24 are all divisible by 3, so each person could take one third.

Finally, a potential answer: They found 25 eggs, and each had 2 eggs for breakfast. Jo got 11 eggs, Harry got 8 eggs, and Greg got 6 eggs.

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