The Gardener's Odd Request

 

Story:

A kind-hearted gardener was known for his love of apples. One day, he harvested a basket full of perfect, ripe red apples. As he was admiring his bounty, three children approached him, each asking for some apples.

The gardener, being fair, told the first child, "You can have half of the apples in my basket, plus one more apple."

He then turned to the second child and said, "You can have half of the apples that are now left in the basket, plus one more apple."

Finally, he smiled at the third child and offered, "You can have half of the apples that are now left, plus one more apple."

After giving apples to all three children as promised, the gardener looked into his basket. To his surprise, the basket was now completely empty.

The Question:

How many apples did the gardener have in his basket at the very beginning?

---

This problem is designed to make you think "too much" by trying to calculate forward, but it's much simpler if you work backward.

---

Solution to The Gardener's Odd Request

This problem is best solved by working backward from the end.

1. When the third child took apples: The gardener gave the third child half of the remaining apples, plus one more. After this, the basket was empty.

   • If taking "half + one" left nothing, then before taking that "one more," there must have been exactly 1 apple (which was "half").
   • So, that "one more" apple must have been the only apple left after giving away half. This means before the third child took any, there were 2 apples in the basket. (Half of 2 is 1; 1 + 1 = 2. So 2 apples given, 0 left).

2. When the second child took apples: The second child took half of the apples left, plus one more. After this, 2 apples remained (as calculated in step 1).

   • Before the second child took that "one more" apple, there must have been $2+1=3$ apples.
   • These 3 apples represented half of what was in the basket before the second child took any.
   • Therefore, before the second child took apples, there were $3\times 2=6$ apples in the basket. (Half of 6 is 3; 3 + 1 = 4 apples given, 2 left).

3. When the first child took apples: The first child took half of the apples initially in the basket, plus one more. After this, 6 apples remained (as calculated in step 2).

   • Before the first child took that "one more" apple, there must have been $6+1=7$ apples.
   • These 7 apples represented half of the original number of apples in the basket.
   • Therefore, the gardener had $7\times 2=14$ apples in his basket at the very beginning. (Half of 14 is 7; 7 + 1 = 8 apples given, 6 left).

---

Answer: The gardener had 14 apples in his basket at the very beginning.