# The banana length puzzle

This is a puzzle from How to bake π by Eugenia Cheng, if you haven’t checked it out, make sure to do so. Here we go…

A rope over the top of a fence has the same length on each side, and weighs one-third of a pound per foot. On one end hangs a monkey holding a banana, and on the other end a weight equal to the weight of the monkey. The banana weighs 2 ounces per inch. The length of the rope in feet is the same as the age of the monkey, and the weight of the monkey in ounces is as much as the monkey’s mother. The combined age of the monkey and its mother is 30 years. Half the weight of the monkey plus the weight of the banana is a quarter of the sum of the weights of the rope and the weight. The monkey’s mother is as half as old as the monkey will be when it’s three times as old as its mother was when she was half as the monkey will be when it’s three times as old as its mother was when she was half as old as the monkey will be when it’s as old as its mother will be when she’s four times as old as the monkey was when it was twice as old as its mother was when she was a third as old as the monkey was when it was as old as its mother was when she was three times as old as the monkey was when it was a quarter as old as it is now. How long is the banana?

Trust me, it’s harder to type out the riddle than to actually solve it. The challenge was to do it mentally, but I found it too hard and I went on paper. The riddle looks long-winded (it is) and complicated, but it’s easier than its length. So basically, to summarize the italicized text, if the monkey’s age is ma and the mother’s age is mma, then

ma = 2/3 mma

or, the mother’s age is a half older than her son/daughter. With easy calculations, the monkey is 12 years old, 18 ounces heavy. The rope is 12 feet long, which is 4 pounds. That means that 6 + the banana’s weight in ounces = 22, meaning the banana is 16 ounces, thereby 8 inches.

BAM!

Disclaimer: I’m not sure if this is 100% accurate, but it’s good. 🙂 Good riddle, Eugenia Cheng!