## Results for My Favorite Game

Thanks to everyone who participated in the online version of my favorite game. Thanks, especially, to those people who helped to share it via Twitter, Facebook, and other blogs.

Though I am posting the results today, I will continue to leave the form online. Though I had only planned to let the contest last one week, entries continue to roll in, and I see no reason to forbid people from playing. From time to time, I’ll update this page… such updates will occur at the intersection of two events: when enough entries warrant an update, and when the muse hits me.

Without further adieu, here are the results.

With 1,042 entries divided into groups of 100, there were 10 complete games played. The charts below show the results for each game. (Sorry if they’re a little hard to read. Click on the images to view them full-size in a separate window. There are two images below — games 1-5 are shown in the top image, and games 6-10 are shown in the bottom image.)

The graphs above do not reflect all 100 entries for each game. In each game, many numbers greater than 35 were chosen. However, 84.1% of all selected entries were 35 or less.

The winning numbers, respectively, were 3, 3, 20, 4, 7, 2, 16, 4, 4, 2.

Congratulations to Loïc Grobol of Chécy, France! Loïc was the winner of the 4th game, and his name was randomly selected as the overall winner of the signed copy of Math Jokes 4 Mathy Folks and the specially-designed, one-of-a-kind random number generator. (There is beautiful symmetry that Loïc chose the number 4, that he was the winner of the 4th game, and that the prize has a 4 in the title.)

As shown in the graphs above:

• On average, the number 1 was chosen by 10.4% of entrants. (Edward Early of St. Edward’s College said, “I use that [game] as a bonus question on a test in every class I teach. I believe 1 has never been the winning number, even in a class with only 6 students.” I have played this game over 100 times with various size groups. In my experience, the number 1 has only won twice… by the same woman, who — in a show of incredible bravado — chose the number in consecutive rounds of the game.)
• The number 2 won twice, but it was chosen rather infrequently — less than 1/3 as often as 1. My suspicion is that people figure if you’re gonna go big, go REALLY big… why choose 2 when you can choose 1?
• Besides 1, the number most often chosen was 17, which was selected 5.5% of the time. This seems to corroborate numerous studies that found 17 to be the most commonly selected random number.
• The number 151 was the greatest number chosen more than once.

The chart below shows the frequency of the top nine guesses.

Among the most interesting entries were 666, 1012, 1337, 53,479, and 3,010,994.

The most amusing entry was 10, with the accompanying note, “I’ll choose the base later.”

And you may be wondering… if all 1,000 entries were considered as just one game, what number would have won? That distinction would have gone to 32.

By running this contest, I learned about two interesting uses of this game in classrooms.

Edward Early said that he uses the following version as a bonus question.

Write a positive integer in the blank: _______

How this will be graded: The least positive integer that is submitted by exactly one person will be worth 5 points. The next-smallest will be worth 4 points, and the next-smallest after that will be worth 3 points. All other positive integers submitted by exactly one person will be worth 2 points. Positive integers submitted by more than one person will be worth 1 point. Anything other than a positive integer will receive no credit. Do not ask me to explain this question.

Not surprisingly, with these modifications comes a change in strategy — Edward said that some students choose a large random number, just to ensure they receive 2 points.

Matt Skoss of Possum Educational Services and the Northern Territory Dept of Education and Training said that he’s used this game for years with his kids at school.

Pick the lowest prime number, composite number, surd, cube number or triangular number, etc., depending upon what I’d like the kids to think about.

What an excellent use of a simple game!

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• 1. Rich Morrow  |  December 7, 2011 at 1:56 am

2, 3 or 4 either won or could have won each time. Strange. Now what would happen if people saw these results and played (essentially) the same people again?? I am reminded of the prisoner’s dilemma programming game.

Rich

• 2. venneblock  |  December 7, 2011 at 3:28 pm

Rich, see the post https://mathjokes4mathyfolks.wordpress.com/2011/04/26/analyzing-my-favorite-game/, where I show the results from this game played for 5 rounds by the same group of people. What happened there is that people chose lower numbers in consecutive rounds, until the group decided that the numbers were getting too low, and then the numbers popped back up again.

• 3. Rich Morrow  |  December 15, 2011 at 1:02 pm

I wonder if there is some kind of a limit pattern on this as the number of rounds gets very large. It just seems that 1 and 17 could not continue as the most picked numbers. It also seems that more people (such as Dave) would begin picking 2, 3 and 4 pushing the winning numbers up. Just hypothesizing here.

• 4. Evan  |  December 7, 2011 at 2:15 am

Now that the game is finished, should we discuss optimal strategy?
It seems to me that in this game, with n players, you should avoid choosing a number larger than n/2 (they *could* win, but only if there is some number smaller than n/2 that no one chose).
Also, there appears to be no equilibrium in pure strategies… the only equilibrium strategies (strategies so that everyone is happy with how they played) must involve randomisation.

However, given the counter-intuitive nature of mixed strategy equilibrium, finding the equilibrium doesn’t really help decide what you should play.

• 5. venneblock  |  December 7, 2011 at 3:31 pm

I’ve played this game with many groups, Evan, and your theory about n/2 sounds good. But I have some data to give an even better estimate for the number to choose. In all the games here, the winning number was never greater than n/5. See the post at https://mathjokes4mathyfolks.wordpress.com/2011/04/26/analyzing-my-favorite-game/; in those games with 32 people, the winning number was never greater than n/7. And I played 5 rounds with a group of 170 people once, and the winning number with that group was never greater than n/16.

• 6. Dave  |  December 14, 2011 at 3:10 pm

I would look at what number(s) I could have played that would have won:

Game 1) Not possible
Game 2) Not possible
Game 3) 4
Game 4) 2, 3
Game 5) 3, 5
Game 6) 3, 5 (Is this accidentally a graph of the Game 5 data?)
Game 7) 4
Game 8) Not possible
Game 9) 3
Game 10) Not possible

Based on this alone, I would go with “3” if I were playing in Game 11.

• 7. venneblock  |  December 15, 2011 at 10:42 am

Holy schnikeys, Dave — you’re right! It was the wrong graph for Game 6. My apologies! That has now been updated, as well as other info in the post. (The winner for Game 6 was actually 2, not 7 as previously stated.)

So using your theory for the updated Game 6, there was no number that would won. Still, I think your choice of 3 is a good one.

• 8. Rich Morrow  |  December 15, 2011 at 1:03 pm

If two people are like Dave, I guess I would go for 4. Is this one of those situations where you get in trouble by either underthinking or overthinking others?

• 9. Bon Crowder  |  January 3, 2012 at 5:45 pm

Wow.

I’m going to play this with my family.

Wow.

• 10. Outlier Babe  |  January 12, 2012 at 12:10 am

Fascinating looking at something like this as a math-phobe. First, wondering why anyone would want to play this. Then, whenever in my past a Math teacher said something like: “There is beautiful symmetry that Loïc chose the number 4, that he was the winner of the 4th game, and that the prize has a 4 in the title.”–I would think “It’s just numbers, and there is no beauty in simple coincidences!” And yet, I find the same fascination with equivalent coincidences in language, or in patterns (yeah, I wasn’t math-phobic in Geometry). So, I get that you-all are seeking patterns. For us math-phobes, it is just too bad that numbers get in the way. It also sometimes seems like you math-a-holics are just as happy when no patterns emerge as when they do. It’s like you’re all Scrooge McDucks, giddy from sifting loose piles of digits through your fingers. (oops–was that a little harsh? i’m just bitter ’cause I flunked Trig. twice. dumb*ss.)

• 11. Active Math Game: Rock | DeniseGaskins.com  |  October 5, 2015 at 7:35 am

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