You'll want to give up on this week's puzzle. Don't.
The key to unraveling this week's puzzle is the key to cracking most good brain teasers: If you want to solve it, you're going to have to get organized.
I'm going to come right out and say it: Our puzzle this week involves math. I see some of you clicking away already. To you I say: Don't be intimidated. Yes, you will need algebra to solve this puzzle, but the most difficult aspect of this week's brain teaser, in my opinion, doesn't boil down to math. It boils down to organization.
A few weeks ago, somebody asked in the comments if a math problem could really be considered a "puzzle." (I can't find the comment now, but I think it was made in reference to the posted solution for The Logician's Children.) The answer to this question is, of course, yes. Martin Gardner, one of the most prolific puzzle-posers of the 20th century, built his career, in large part, on mathematical puzzles. A lot can be said about why math problems make for excellent puzzles, but the simplest explanation I can provide is that a good math problem demands to be solved by means of a carefully organized approach.
Entire books could be written about this last point. In fact, they have been: How to Solve It, the classic text by mathematician G. Pólya, is considered by many to be the definitive guide to mathematical problem solving. In his book, Pólya outlined a list of best practices for confronting puzzles of the mathematical variety. His approach, originally written in 1945, has since been distilled into a plan of attack by Herman Gordon, associate professor of cell biology and anatomy at the University of Arizona, and instructor of The Art of Scientific Discovery, a course designed to hone students' skills at solving the kinds of problems one encounters in medicine and scientific research. Gordon's guide walks us through the way a good mathematician (and a good problem solver) thinks. "Even though the topic is logic," he writes, "the discovery and solution of mathematical problems involves induction and heuristic thinking:
1. Understand the problem
- What is the unknown?
- What are the data?
- What is the condition?
- Can the problem be solved?
- What can you or need you assume?
- What shouldn't you assume?
- Have you made subconscious assumptions?
3. Devising a plan of attack
- Have you seen this or a related problem before?
- Have you seen a similar unknown before?
- Can you restate the problem?
- If you can't solve this problem, can you solve a similar or simpler problem?
- Are you sure of the solution? Can you see it at a glance?
- Did you use all the data? The whole condition?
- Can you get the same solution another way?
- Are there other valid solutions?
- Can you apply the solution or method to another problem?
- Was this a satisfying problem to solve?
If you ever find yourself struggling with a Sunday Puzzle, or any puzzle for that matter, look to these guidelines. Print them out. Stick them above your desk. I guarantee you will find this guide – or, at the very least, some part of it – helpful when puzzling your way through a good brain teaser, be it a math problem, a logic puzzle from our weekly column, or a line of scientific inquiry.
Becoming a better problem solver requires practice. You need to be organized. And you need to think about your thinking. With all that in mind, here is this week's puzzle.
A monkey and his uncle are suspended at equal distances from the floor at opposite ends of a rope which passes through a pulley. The rope weighs four ounces per foot. The weight of the monkey in pounds equals the age of the monkey's uncle in years. The age of the uncle plus that of the monkey equals four years. The uncle is twice as old as the monkey was when the uncle was half as old as the monkey will be when the monkey is three times as old as the uncle was when the uncle was three times as old as the monkey. The weight of the rope plus the weight of the monkey's uncle is one-half again as much as the difference between the weight of the monkey and that of the uncle plus the weight of the monkey.
How long is the rope?
How old is the monkey?
This week's puzzle was submitted by Reid (a mathematician) in retribution, he says, for last week's puzzle, which – despite appearances – actually required no mathematical skills whatsoever to solve.
We'll be back next week with the solution – and a new puzzle! Got a great brainteaser, original or otherwise, that you'd like to see featured? E-mail me with your recommendations. (Be sure to include "Sunday Puzzle" in the subject line.)
Art by Jim Cooke
Last week, I asked you to deduce the next number (or numbers) in this sequence, with the caveat that the mathematically inclined tend to struggle with this puzzle:
I was overwhelmed by your response. A few hundred of you e-mailed me correct answers.
Several of you told me that the hint about mathematicians struggling with this problem was too big a hint, so for that I apologize. On the other hand, several mathematicians (including Reid, who submitted this week's puzzle), mentioned they probably never would have solved it without that tip. ¯\_(ツ)_/¯
On the off chance that there are people reading this who have yet to solve the sequence, but would still like one last big hint, I'm going to present this week's solution a little differently. Rather than giving you the solution outright, I'm going to link to the Wikipedia page describing the sequence, which also provides some of the history (and, believe it or not, the math!) behind it.
The Wikipedia entry in question is on The "Look-And-Say" Sequence.