Sam Loyd, Weight Problems, and Exercises

Welcome to this week’s Math Munch!

Chess master, puzzlist, and recreational mathematician Sam Loyd. GREAT mustache.

Chess composer, puzzlist, and recreational mathematician Sam Loyd. GREAT mustache.

First up, remember Sam Loyd?  (We’ve featured him twice before.)   He was an american chess player and recreational mathematician who lived from 1841-1911.  He was also a chess composer, someone who writes endgame strategies and chess puzzles.  In fact, he wrote all sorts of puzzles, which his son published in a book called Sam Loyd’s Cyclopedia of 5000 Puzzles, Tricks, and Conundrums.  (That link will take you to a scan of all 385 pages!)  By the way, those 5000 puzzles are only about half of the ones he wrote in his lifetime.  It’s no wonder Martin Gardner called him “America’s greatest puzzler.”  An interesting anecdote: Sam Loyd claimed until his death to have invented the 15 puzzle, but in fact he did not.  The actual inventor was Noyes Chapman, the Postmaster of Canastota, NY.

I wanted to show you some of Sam’s “Puzzling Scales” problems.  Why don’t you stop reading now and just solve them both?

Puzzling Scales 1 Puzzling Scales 2
Torque puzzle 1

These different weights balance because of the torque they apply

There are lots of puzzles like this, based on different weights balancing with each other.  A friend sent me this page of weight puzzles based on the idea of torque.  The farther out an object is placed, the more torque it applies to the balance, so it’s possible for a 1 pound weight to balance a 2 pound weight if you set them at the right distances.  The distance and wight multiply to give the torque applied.

These problems come from a massive bank of puzzles over on Erich’s Puzzle Palace.  If you like, you can also play this torque game I found.

Torque puzzle 1

Place 1 through 5 to balance the weights.

Torque puzzle 2

Place 1 through 6 to balance the weights.

I love problems like this, but I started to wonder, “what if the scales don’t balance?  Maybe you could make a puzzle out of that.”  I did exactly that, creating a series of imbalance puzzles.  Your job is to order the shapes by weight.  They start out easy, but there are some tricky ones.  I especially like #6.

In each case, order the three objects by weight.

Imbalance 1 Imbalance 2 Imbalance 3

I’m also hosting an imbalance puzzle-writing contest.  My two favorite puzzlists will win a print of their choosing from my Stars of the Mind’s Sky series of mathematical art.  You should try your hand at writing one.  Just email it to Lost in Recursion.

Finally – we all love great problems and puzzles, but skill building is an important aspect of mathematics as well, and exercises help us build skill.  Exercises are often dull, but I found a website with some exercises I quite like, and I wanted to share them with you.  Check out the Coffee Break section over on

Detention Dash

Detention Dash

Find the Primes

Find the Primes

Odd One Out

Odd One Out

Detention Dash, for example, is just a timed multiplication chart, but typing the answers in on my computer really made me feel some of the patterns in the numbers.  You should try it.  Odd One Out also keeps you on your toes and makes you think about different kinds of numbers.  I find them surprisingly fun.  I hope you agree.

Bon appetit!

8 responses »

  1. I love the torque puzzles. I once took a math tutoring class from Persis Herold, who ran the Math Center in her house in D.C. She told us that seesaws are so important for children to play on because they give the children an understanding of balance, etc. that helps them in math and science. I always encourage my students to ride seesaws if they are given the opportunity. This whole concept of torque would be so much more intuitive for kids who have experienced the ups and downs of a seesaw! 🙂

  2. The name Sam Loyd sounds familiar but after reading the article about him I realized that I didnt know who he was at all. Im not a fan of chess but puzzles are always fun when you understand how to solve them correctly. Torque is something that Ive never heard about when dealing with scales and balancing. In the lesson that I’m learning right now I’m using a form of balance that helps me solve equations with variables on both sides by simply drawing a line down the wherever the equal sign is. I feel that excercises are good for the brain when your first period of the day is math; they wake up your brain! The excercises in this post seem challenging and fun, but will give a brain a good workout!

  3. What I noticed with Sam Lloyd’s puzzling scale is that the shell is equal to the block and eight marbles, so on the top scale each block is worth two marbles. On the last scale you have to add eight marbles and 2 marbles , because the block is worth two marbles and the answer is 2. During the time i was doing this puzzle it was very confusing to me at first then i got the hang of it and figured it out. I liked this puzzle because it was very interesting.

  4. Pingback: A Periodic Table, Linkages, and Dance Squared | Math Munch

  5. Pingback: 131- weight puzzles | 180 Days of Math at Mesa

  6. Wow! This is one of my favorite math munch posts! I love how math is made into less of a chore, or a ‘problem’, and more of a game! I would love to solve more problems like this.

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