FiveThirtyEight's The Riddler


What is the Riddler?

The riddler is a weekly math, logic and probability problem hosted by fivethirtyeight.com, a statistically oriented politics, sports and culture blog.


If the answer is particularly cool or my solution can be extended, made interactable or generalized then I'll usually write it up. I've also created and submitted some problems that have been featured.

Submitted Problems


Barron Squares

This is a mathmatical object I made and obviously named after myself

It is a square matrix such that each row's leftmost element times its rightmost element will equal the two inner numbers (in the example below, the top row has 6*7 = 42). Similarly, each column's uppermost element times the column's lowermost element equals the middle digits (in the example, the right column is 7*8 = 56), as read top to bottom.

6 4 2 7
4 2 0 5
8 4 8 6
8 6 4 8
4 x 4 Barron Square

This was then expanded on to be a 8x8 with two digit rows/columns being multiplied to get four digit products


6 2 4 5 8 8 7 4
6 1 2 9 2 8 4 8
1 1 0 4 6 2 4 2
7 6 4 4 8 4 5 9
8 3 6 8 0 6 8 2
2 8 2 4 6 4 8 8
2 7 1 7 8 2 6 6
7 8 1 6 3 8 2 1
8 x 8 Barron Square

We ended up finding all 1444 4x4 Barrons Squares, several hundred 8x8 and even a few 16x16 Barron Squares. This was featured as a Riddler Express for FiveThirtyEight.




A Geometric Puzzle

This is a problem about finding geometric subshapes and summing their areas

The largest circle below has a radius of 10, the medium circle has a radius of 5 and the small, orange circle has a radius of 2. The orange circle crawls counterclockwise along the edge of the largest circle until it meets the medium circle, at which point it crawls up along the edge of the medium circle until it reaches the crest. What is the area of the shaded orange region in the right image?

Find the area of the orange section



Unique Tile Snips

This is a problem about coloring a square so if a 2x2 section is snipped out you know where it was originally located


You are given an empty 4-by-4 square and one marker. You can color in the individual squares or leave them untouched. After you color as many or as few squares as you’d like, I will secretly cut out a 2-by-2 piece of it and then show it to you without rotating it. You then have to tell me where it was (e.g., “top middle” or “bottom right,” etc.) in the original 4-by-4 square.

Can you design a square for which you’ll always know where the piece came from?

150 of the 6188 solutions

The solution can be found here




7 Segment Display

This problem is about reducing ambiguity in seven-display segments

Given a two-character, seven-segment display, like you might find on a microwave clock, how many numbers can you make that are not ambiguous if the display happens to be upside down?

For example, the number 81 on that display would not fit this criterion — if the display were upside down it’d appear like 18. The number 71, however, would be OK. It’d appear something like 1L — not a number.

a seven-segment display

The solution can be found here

Problem Solutions



Putting Sharp Objects into Round Ones

This problem was about creating triangles in a unit circle and determining if the circle's center was enclosed.

Choose three points on a circle at random and connect them to form a triangle. What is the probability that the center of the circle is contained in that triangle?

This led to the following visualization:

Results from 2d simulation

This was followed up by the 3d case:

Choose four points at random (independently and uniformly distributed) on the surface of a sphere. What is the probability that the tetrahedron defined by those four points contains the center of the sphere?


Results from 3d simulation

The probability that the shapes would enclose the center resulted in surprisingly nice 25% and 12.5%




Gerrymandering

This problem was about taking a state map, show below, and gerrymandering it as much as you could.

Below is a rough approximation of Colorado’s voter preferences, based on county-level results from the 2012 presidential election, in a 14-by-10 grid. Colorado has seven districts, so each would have 20 voters in this model. What is the most districts that the Red Party could win if you get to draw the districts with the same rules as above? What about the Blue Party? (Assume ties within a district are considered wins for the party of your choice.)

Initial map with no districts drawn

We used a randomized search that would try to slightly move districts, check continuity and then check the voting results. The method resulted in uniquely shaped maps that optimized each side's winningness.

Districts move their boundaries each iteration

Map with districts drawn optimized for blue. Blue wins 5/7 districts after 100,000 iterations



Creating Crossword Puzzles

This problem is about making crossword puzzles

Crossword puzzle grids typically obey a few rules and conventions.
  • They are 15-by-15.
  • They are rotationally symmetric — that is, if you turn the grid upside down it appears exactly the same.
  • All the words — that is, all the horizontal and vertical sequences of white squares — must be at least three letters long. All the letters must appear in an “across” word and a “down” word.
  • The grid must be entirely connected — that is, there can be no “islands” of white squares separated from the rest by black squares.
How many such crossword grids are there?


While I didn't find them all I was able to generate a few thousand. Turns out none of the entrants were able to find them all






Others:

Game Theory on a Number Line

The Eccentric Billionaire and the Banker

A Tale of Two Endpoints

Elf Playlist

Creating Crossword Grids

Sum to 15 Card Game

A FiveThirtyEight Spelling Bee

A Circular Conundrum

Matching Game

A Classic Construction Problem

An Amphibious Enigma

Bright Lights, Big City

The Purge

A Painting Puzzle

Pick a Card, Any Card

A Puzzle About Domestic Boundaries

An Impromptu Gambling Problem

Can You Survive This Deadly Board Game?

Count Von Count

Jems

Space Race

Which Geyser Gushes First?

What’s The Best Way To Drop A Smartphone?