Which Billiard Ball Is Rigged?

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-size 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 have a favorite puzzle collecting dust in your attic, find me on Twitter.

## Riddler Express

From Corey Grodner, balance a bunch of billiard balls:

You have 12 billiard balls. To the naked eye, they all look identical, and in your hand, they all feel identical. One of the balls, however, is slightly heavier or slightly lighter than the others, but you don’t know which ball or whether it is heavier or lighter.

You do, however, have a balance scale. You can place any equal number of balls on each side of the scale, and the scale will tilt if one side differs in weight. (Note: There is no use in weighing different numbers of balls against each other — the weight difference is so slight that if the scale has more balls on one side, that side will always be heavier.) However, you can only use the scale three times.

How can you determine which ball is different, and whether it is heavier or lighter?

## Riddler Classic

From Dave Moran, in which we revisit his classic Riddler technicoloring game:

Almost three years ago, we first introduced in this column a two-player map-coloring game played with writing utensils and a piece of paper. It works like this: Call the two players Allison and Bob. On each turn, Allison draws a simple closed curve on a piece of paper. Bob then colors the interior of the “country” that curve creates with one of his many crayons. If the new country borders any existing countries, Bob must color the new country with a color different from the ones he used for the bordering countries.

For example, the game might begin with Allison creating Country 1, Bob coloring it green, Allison creating Country 2, Bob coloring it pink, and so on.

In this edition of the game, suppose Allison wins when she forces Bob to use a seventh color. How many countries must Allison draw to win?

For context, Allison won the original game a few years ago when she forced Bob to use a sixth color. In that instance, she needed to draw eight countries.

Extra credit: Is there an upper limit on the number of colors Bob can be forced to use? If so, what is it? If not, why not?

## Solution to last week’s Riddler Express

Congratulations to 👏 Amanda Pierson 👏 of Washington, D.C., winner of last week’s Riddler Express!

Last week, the Puzzle Party National Committee was organizing a series of debates for the 20 candidates running for its presidential nomination. Twenty was too many to fit on a stage at once, so in each round of debate the PPNC would separate the candidates into two panels of 10. Candidates, by PPNC rule, were only allowed to criticize those candidates with whom they shared a stage. But the PPNC wanted to provide ample opportunities for such shade-throwing. What was the minimum number of rounds of bifurcated 10-candidate debates such that each candidate got an opportunity to personally attack every other candidate?

The minimum number of rounds was three.

For example, in the first debate, the panels are set like this:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10

11, 12, 13, 14, 15, 16, 17, 18, 19, 20

In the second debate, like this:

1, 3, 5, 7, 9, 11, 13, 15, 17, 19

2, 4, 6, 8, 10, 12, 14, 16, 18, 20

And in the third debate, like this:

1, 3, 5, 7, 9, 12, 14, 16, 18, 20

2, 4, 6, 8, 10, 11, 13, 15, 17, 19

Scott Wickham offered a helpful explanation for how to arrive at such an arrangement: Split the candidates up into what effectively becomes four teams of five people. They’ll move together as a unit. You can then think of the debates as a round-robin tournament, where each “team” plays the other three teams once — therefore enabling candidates to see the other 15 people on stage over the course of three debates, as well as the four people they are paired with in every debate.

## Solution to last week’s Riddler Classic

Congratulations to 👏 Orestis Papaioannou 👏 of Nicosia, Cyprus, winner of last week’s Riddler Classic!

Last week, you pondered the constructive possibilities of a pond you owned (with a radius of 1 yard) and your large collection of thin boards (each 1 yard long). Specifically, you wondered: How many boards were necessary in order to cover the center point of the pond with a board? To avoid falling in the water, the ends of each board needed to lie on either the banks of the pond or on another previously placed board.

You could cover the center point with as few as 12 boards.

That arrangement looks like the image below, as constructed by solver Tim Black, who also wrote a lovely blog post about how his brain and his computer combined forces to arrive at this solution. The main idea is to build a sort of scaffolding inward from the banks of the pond that is both large enough to eventually build into the center but sparse enough to avoid using too many boards. One helpful way to approach this, as with many Riddlers, is to work backwards — start with the board that will reach the center point and build your scaffolding out from there.

Solver Zach Wissner-Gross tried ever so hard to construct an 11-board arrangement, but demonstrated in the animation below that 11 boards just aren’t quite enough.

And why stop with just this one pond? This puzzle’s submitter, Erich Friedman, has assembled a collection of efficient center-covering board arrangements for ponds of various sizes — and not just circular ponds but triangular, square and pentagonal ponds as well.

Have a great weekend. If the weather’s nice, you can find me down at the local pentagonal swimming hole, a pile of boards under my arm.

## Want more riddles?

Well, aren’t you lucky? There’s a whole book full of the best puzzles from this column and some never-before-seen head-scratchers. It’s called “The Riddler,” and it’s in stores now!

## Want to submit a riddle?

Email me at oliver.roeder@fivethirtyeight.com.

## Footnotes

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

Oliver Roeder was a senior writer for FiveThirtyEight.