# Making Arrows

This is an interesting problem from 180 BC China.

“In one day, a person can make 30 arrows or fletch [put the feathers on] 20 arrows.  How many arrows can this person both make and fletch in a day?”

It turns out the solution to this problem led me into the history of numerator/denominator (aka common) fractions, a subject I had been finding difficult to track down.

See Making Arrows for a solution.

# Bailing Water Problem

This is a straight-forward problem from Five Hundred Mathematical Challenges.

“A boat has sprung a leak.  Water is coming in at a uniform rate and some has already accumulated when the leak is detected.  At this point, 12 men of equal skill can pump the boat dry in 3 hours, while 5 men require 10 hours.  How many men are needed to pump it dry in 2 hours?”

See the Bailing Water Problem for solution.

# Refabulating Widgets

This is a work problem from Geoffrey Mott-Smith from 1954.

“ ‘If a man can do a job in one day, how long will it take two men to do the job?’

No book of puzzles, I take it, is complete without such a question. I will not blame the reader in the least if he hastily turns the page, for I, too, was annoyed by “If a man” conundrums in my schooldays. Besides, the answer in the back of the book was always wrong. Everybody knows it will take the two men two days to do the job, because they will talk about women and the weather, they will argue about how the job is to be done, they will negotiate as to which is to do it. In schoolbooks the masons and bricklayers are not men, they are robots.

Strictly on the understanding that I am really talking about robots, I will put it to you:

If a tinker and his helper can refabulate a widget in 2 days, and if the tinker working with the apprentice instead would take 3 days, while the helper and the apprentice would take 6 days to do the job, how long would it take each working alone to refabulate the widget?”

See Refabulating Widgets for a solution.

# Alcan Highway Problem

This work problem from Geoffrey Mott-Smith is a little bit tricky.

“An engineer working on the Alcan Highway was heard to say, “At the time I said I could finish this section in a week, I expected to get two more bulldozers for the job. If they had left me what machines I had, I’d have been only a day behind schedule. As it is, they’ve taken away all my machines but one, and I’ll be weeks behind schedule!”

How many weeks?”

See the Alcan Highway Problem for a solution.

# Fibonacci, Chickens, and Proportions

There is the famous chicken and the egg problem: If a chicken and a half can lay an egg and a half in a day and a half, how many eggs can three chickens lay in three days? Fibonacci 800 years ago in his book Liber Abaci (1202 AD) did not have exactly this problem (as far as I could find), but he posed its equivalent. And most likely the problem came even earlier from the Arabs. So we can essentially claim Fibonacci (or the Arabs) as the father of the chicken and egg problem. Here are three of Fibonacci’s actual problems:

1. “Five horses eat 6 sestari of barley in 9 days; it is sought by the same rule how many days will it take ten horses to eat 16 sestari.
2. A certain king sent indeed 30 men to plant trees in a certain plantation where they planted 1000 trees in 9 days, and it is sought how many days it will take for 36 men to plant 4400 trees.
3. Five men eat 4 modia of corn in one month, namely in 30 days. Whence another 7 men seek to know by the same rule how many modia will suffice for the same 30 days.”

By modern standards these problems all involve simple arithmetic to solve. But there are actually some subtleties in mapping the mathematical model to the situation, in which fractions, proportions, ratios, and “direct variation” get swirled into the mix—naturally causing some confusion.

See Fibonacci, Chickens, and Proportions for a solution.