The Reward

In this math puzzle, you need to figure which reward choice is worth more money.

The king of Seedonia was driving along a winding mountain road with his son. The foolish prince was not wearing his seat belt, so when the king had to swerve to avoid hitting an elephant that was strolling across the road, the passenger side door opened and the prince was thrown out. He tumbled to the side of the road and was left hanging by his fingertips over the valley several thousand meters below.

Fortunately, a young peasant girl happened by just in time and pulled the prince to safety. The grateful king invited her home to the palace for dinner and a game of chess.

After the game the king offered the girl a reward of 1,000,000 Seedos for saving the prince. (The Seedo is the unit of currency in Seedonia. It is equal in value to a Euro.)

The girl looked at the chess board and suggested an alternative reward:

“Put one Seedo on the first square of the chess board. Tomorrow, remove the Seedo from the first square and put two Seedos on the second square. The day after tomorrow, remove the two Seedos and put four Seedos on the third square. The next day, put eight Seedos on the fourth square, and so on each day. On the 64th day, I will leave with whatever amount is on last square of the board.”

The king, who was not a very good math student, thought this was a very modest request. He agreed to the girl’s wishes, thinking he would spend quite a bit less than his original offer.

How much money did the girl end up with? Was it more or less than 1,000,000 Seedos? Make a guess before you try to calculate it.

Hint: This problem is very similar to the one we presented in How Thick Does It Get? [1].

Course:

Math [2]

Result/Solution(s)

The king went broke long before the 64th day. The girl took over the palace and declared herself Queen of Seedonia. She ended up with

18,446,744,073,709,551,616 Seedos.

If the king had looked at the pattern that was emerging, he would have seen that he was headed for big trouble:

1, 2, 4, 8, 16, 32 . . .

These are the powers of 2. The sequence may be written:

2⁰, 2¹, 2², 2³, 2⁴, 2⁵. . .

After 64 days, we have

2⁶⁴ = 18,446,744,073,709,551,616 Seedos.

Actually, the girl was being modest in her request. In another version of this puzzle, the money is left on each square instead of being removed. In this case the total would equal:

2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁵ + . . . + 2⁶² + 2⁶³ + 2⁶⁴

The moral of the story is: Beware of exponential deals.

Math Puzzle [3]