Years ago a blacksmith put a set of new shoes on a horse owned by a wealthy but stingy rancher. When the job was completed, the blacksmith charged the rancher ten dollars for the job. The rancher claimed the price was too high and refused to pay.

"Very well," the blacksmith told him, "I'll make a deal with you. Each horseshoe has eight nails. There are four shoes on the horse. Four eights are thirty-two. I'll charge you one cent for the first nail, two for the next, four cents for the next, and so on for all thirty-two nails."

The rancher agreed at once and reached for his wallet. How much will he have to pay?

This puzzle is from "Puzzle it out" book by Richard and Churchill.

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## 4 comments:

OK, this was NOT a good deal for the rancher, and I'm betting there was not enough money in his wallet.

If he paid 1 cent for the first nail, 2 for the second, 4 for the next, 8 for the fourth, etc., then he was paying $5.12 for the 10th, over $5000 for the 20th, and the 32nd nail would be over $21million. In total = 42,949,672.95.

$10 is a bargain in comparison.

Doggone, there is some symbol for this and I cannot remember it. It's not "factorial" such as "2!" which is multiplying the integers; this is a sum of products .... 1+2+4+8+16+32 etc. not gonna do it today. I bet Kim's right.

The same arithmetic comes up in the "Martingale" system of gambling, continually placing doubled bets after any loss, in a game where the odds are nearly even such as roulette (red/black) or basic craps (pass/nopass). (Not a safe system people; pretty soon you lose anyway, sooner than you think, either from the table limits or from your empty purse.) (See martingale at wikipedia, there's some math there too.)

Tom

It is great to see how we all approach these puzzles differently. Some have more formal mathematical education and jump to the familiar formulas, some use brilliant analogies, life examples or intuition.

Kim, of course, is right. Even so it is hard to believe one cent could grow into millions so quickly. We play with powers of two here.

Rancher needs to pay:

Nail - Payment

1st - 1 cent

2nd - 2 cents

3rd - 4 cents (2^2)

4th - 8 cents (2^3)

5th - 16 cents (2^4)

................................

Nth - 2^(N-1) cents

so, for N=32, we'll have 2^ 31 = 2,147,483,648

cents = $2,147,484

the words Tom was searching for a Geometric Progression: every number is larger than a previous number by the same multiplier. In our case, it is every next nail is twice more expensive than previous.

There is also a formula on how to compute a sum of elements of this progression, that will be sum paid for all the nails. But even before we do this, it is obvious that it will be much more expensive than $10.

So, each nail costs the sum of all the prior ones (plus one cent). So if we know the Nth one is 2^(N-1) then we know the sum of all of them up to and including N is 2* 2^(n-1) - 1

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