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Can You Do Math Without Numbers?

Welcome to The Riddler. Every week, I offer up problems related to the things we hold dear around here: math, logic and probability. There are two types: Riddler Express for those of you who want something bite-sized and Riddler Classic for those of you in the slow-puzzle movement. Submit a correct answer for either,1 and you may get a shoutout in next week’s column. If you need a hint, or if you have a favorite puzzle collecting dust in your attic, find me on Twitter.

Riddler Express

From Diarmuid Early, a shapely puzzle first featured in the collection of math problems he co-authored. “I’ve seen this one drive many people crazy!” Early told me:

You are handed the strange shape shown below. Can you cut it into three pieces and reassemble them to form a square?

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Riddler Classic

Also from Early, a starry division problem. “I love this one, because it’s hard to believe at first that it could have a unique solution or that you could find it without a computer search — but it does, and you can,” he said:

In the long division below, each asterisk represents a whole number — any digit from 0 to 9. Reconstruct all the calculations, given that there is no remainder.

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Solution to last week’s Riddler Express

Congratulations to 👏 Nick Arnold 👏 of Avondale, Pennsylvania, winner of last week’s Express puzzle!

You are handed a piece of paper containing a 13-by-13 square, which is divided by lines into a grid, and you must cut it up into some smaller square pieces. If you are only allowed to cut along the lines, what is the smallest number of squares you can divide this larger square into?

While the vast majority of answers submitted were 12, the smallest number of squares is actually 11. Here’s how it looks:

Laurent Lessard turned to integer programming to solve the problem, then illustrated the smallest number of sub-squares you could make out of squares that are smaller or larger than 13-by-13:

More generally, this problem is known as Mrs. Perkin’s Quilt, and it has been the subject of some serious mathematical study. The solutions for larger squares are known for many of these “quilts,” but no general solution for a square of side length N is known. Get to work, Riddler Nation!

Solution to last week’s Riddler Classic

Congratulations to 👏 Mike Strong 👏 of Mechanicsburg, Pennsylvania, winner of last week’s Classic puzzle!

The bugle sounds, and 20 horses make their way to the starting gate for the first annual Lucky Derby. Each second, every horse takes one step. Each step is exactly one meter long. But what these horses exhibit in precision, they lack in sense of direction. Most of the time, their steps are forward (toward the finish line) but the rest of the time they are backward (away from the finish line). You know that Horse One goes forward 52 percent of the time, Horse Two 54 percent of the time, Horse Three 56 percent, and so on, up to the favorite filly, Horse Twenty, who steps forward 90 percent of the time. The horses’ steps are taken independently of one another, and the finish line is 200 meters from the starting gate. What are the odds that each horse wins?

Practically speaking, most of these horses have no realistic chance of winning. Horse Twenty, the favorite, wins over 70 percent of the time. Horse Nineteen wins about 20 percent of the time and Horse Eighteen about 5 percent of the time. That doesn’t leave a whole lot of percentage to go around for the other 17 contenders. Horse Seventeen can pull out a win not quite 1 percent of the time, Horse Sixteen about 0.1 percent of the time, and all the horses after that get laughably long odds. Guy Moore calculated Horse One’s winning chances at about 0.000000000000000000000000000000002 percent. So you’re saying there’s a chance!

Most solvers turned to computer simulation to handicap the race, so answers varied a bit — but the outcomes were similar across the board. Justin Brookman, Andrew Zwicky and Jameson O’Reilly, for example, were kind enough to share their code.

But horses don’t win based on statistical formulae or computer code — they’re crowned in the blood, sweat and tears of a racetrack. Or, at least in this case, a virtual racetrack. My favorite solutions came with visual depictions of the race, so here are a few actual runnings of the Lucky Derby! Pour yourself a tall mint julep and enjoy:

Horse Twenty pulls ahead at the halfway mark in this gorgeous animation, and it never surrenders its lead:

From Russell Wang:

This belongs in MoMA:

Horse Twenty goes wire to wire on a calculator:

Want more? You can spectate races administered by Gabriel Pezanoski-Cohen (do click, it’s adorable) and Tyler Barron, plus view a customizable race by Mikolaj Franaszczuk.

Want to submit a riddle?

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  1. Important small print: For you to be eligible, I need to receive your correct answer before 11:59 p.m. EDT on Sunday. Have a great weekend!

Oliver Roeder is a senior writer for FiveThirtyEight.