Are You Really Any Good at Threes?

A couple of days ago at lunch, Irmak told me about Threes!, a phone game written by Asher Vollmer and Greg Wohlwend.

If you haven’t played it, the basic idea of the game is that there’s a grid of numbered cards and you push the cards around (left, right, up, or down) to try to pair matching numbers. If you push a pair of adjacent threes together, you get a six. If you push sixes together, you get a twelve, and so forth. The higher the number on your cards, the higher your score.

Yesterday morning I was enjoying a Threes binge on my train ride downtown. A few times when I made a bad move, I frustratedly pushed cards around at random, to hurry up and get the game over so I could start a new one. Near the end of the ride, using this it's-not-quitting-if-you-don't-hit-'new-game' strategy, I random-swiped my way to a surprisingly high score (around a 900 — I think my intentional high score was about 1,200 at that point). This, I must admit, was a blow to my ego. I’m clearly not as good as I thought I was if I can just swipe around randomly and get a high score.

Wondering just how well anybody would do if they picked all their moves at random, I wrote a quick program to play a thousand games of Threes (the nice web version by Angela Li) using the random-swipe approach. Below, see what score you would expect to get if your strategy were... “less-than-brilliant.” But first, here’s a mesmerizing screen capture of the program in action:

The results

Here's a histogram of the scores of the random-swiper over a thousand games:

Over 1,000 games, the average score came out to 272. That's a bit of a relief: my average is higher than that. However, once in about every 100 games, rando-swiper scores above a 950, so if you really want to be sure you are better, you'll want to beat that. Aside. The peaks in the distribution are due to the fact that scores from individual cards come in powers of three: 3, 9, 27, 81, 243, 729, ...

Here's another way of looking at it (code is here):

Look up the number of games you've played and your high score. If you're in the green region, you can be confident you're better than random. If you're in the yellow region, you're not doing any better than random. If you're in the orange region, look on the bright side: you have an simple strategy for improving your scores!

Update on April 8th, 2014

I was talking with David Smith from Revolution Analytics (@revodavid) yesterday, and the topic of Threes game came up because he also wrote a blog post about it. He mentioned that a friend recommended a "Tetris-like" strategy: move primarily only in one direction (for example, only use left, right and down). I quickly ran a few hundred simulations with the revised strategy, but it turns out that the distribution of scores is statistically indistinguishable (p=0.89 using bootstrap KS test, R script here) from random-swiping in all four directions.

small squareMar 2014
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