I made it out of clay

Robert Nemiroff and Eva Nemiroff ask: are dreidels fair? Spoiler alert: Betteridge’s law of headlines applies here.

The game traditionally played with the dreidel is unfair, as Ben Blatt showed by simulation and Robert Feinerman showed analytically, but this is assuming that all four sides of the top are equally likely to come up when it is spun. The Nemiroffs took this one step further and checked whether the four sides of the dreidel are equally likely to come up.  They took three dreidels and spun them (800, 1000, and 750 times respectively) and showed that these dreidels were unfair even in this more basic sense.

Interestingly, the patterns seem to tell a story about how the dreidels the Nemiroffs used were flawed. I reproduce their Table 1 here (and yes, they had a dreidel with Christmas imagery on it…)

Driedel ג (gimel)/ Santa נ (nun)/ candy cane ש or פ (shin or pei) / tree ה (he) / snowman total spins
Old wooden 109 302 134 255 800
Cheap plastic 311 243 196 250 1000
Santa 52 275 126 297 750

The letters נ (nun) and ה (he) appear opposite each other, as do ג (gimel) and whichever of ש or פ (shin or pei) is used. So what we see here is that:

  • on the “old wooden” dreidel and the “santa” dreidel, two sides opposite each other are preferred – perhaps the dreidel is slightly wider in one direction than the other
  • on the “cheap plastic” dreidel, one side is preferred and the side opposite it is dis-preferred – perhaps the dreidel is slightly heavier on one side or the handle is slightly off-center.

Presumably dreidels are allowed to be so unfair because nobody is playing dreidel for high stakes, so there’s no real incentive to construct the things properly.

After this year, I can always divide my life into triangles

Today is my 33rd birthday. In honor of that, here are some interesting properties of 33.

One from Wikipedia’s list which I like because I have a soft spot for integer partition problems, is that it’s the largest positive integer that cannot be expressed as a sum of different triangular numbers. The others are 2, 5, 8, 12, and 23: see OEIS A053614. There’s an almost-proof of this fact in this compilation of problems from mathematical olympiad selection tests; that compliation cites this review paper of Erdos and Graham on results in combinatorial number theory, but I can’t find the result there! If I make it to 128, it’s the largest number not the sum of distinct squares.

An idea of the proof is as follows: check by enumeration that 34 through 66 can be written as the sum of distinct triangular numbers, where 66 is not used: 34 = 28 + 6, 35 = 28 + 6 + 1, 36 = 36, 37 = 36 + 1, 38 = 28 + 10, …, 66 = 55 + 10 + 1. Then add 66 to each of these to get a way of expressing 67, 68, …, 132 as a sum of distinct triangular numbers – for example 104 = 66 + 38 = 66 + 28 + 10. Add the largest triangular number less than 132 (this turns out to be 120) to each of those decompositions to write each of 133, …, 252 as such a sum. And so on.

Why is this worth singling out from the list? Many of the others include some arbitrary constant, such as:

  • “the sum of the first four positive factorials”
  • “the smallest odd repdigit that is not a prime number” (a “repdigit” is a number that consists of the same digit repeated, so the constant 10 is hiding here; inf act you could argue this is basically a strange way of stating the identity 33 = 3(10+1))

It’s also pretty cool that 33 is a Blum integer – that is, a product of two distinct primes, each of which is congruent to 3 mod 4. (But it’s not the first Blum integer – that’s 21.)

Another property of 33, which is less negative, is that it’s the first member of the first cluster of three semiprimes (33 = 3 x 11, 34 = 2 x 17, 35 = 5 x 7). That is, it’s the first member of this sequence. In OEIS terms, I’d say that being the first member of a sequence, or the last member of a sequence, is more interesting than being just out in the middle of the sequence somewhere.

The semiprime thing appears to have an arbitrary constant of 3. But there are no clusters of four or more consecutive semiprimes – out of four consecutive integers, one is divisible by 4 – so 33 is the first member of the first cluster of semiprimes of maximal length.

Want to know what’s interesting about some number? You could trawl the OEIS or Wikipedia, or you could go to Erich Friedman’s list, which is a bit more selective, only listing one property of each number. In fact both of my interesting properties of 33 appear here – the semiprime one is, for Friedman, a property of 34, “the smallest number with the property that it and its neighbors have the same number of divisors”.

Time zones and election turnout

Another bit of election analysis: When You Don’t Snooze, You Lose: A Natural Experiment on the Effect of Sleep Deprivation on Voter Turnout and Election Outcomes, working paper by John B. Holbein and Jerome P. Schafer.

People just to the east of a time zone boundary sleep 20 minutes less than those on the west side of the time zone boundary. (This is based on the American Time Use Survey.) This depresses voter turnout, which, in a US setting, moves election results to the right. (Anecdote is not data, but this year I voted early one morning in the week before Election Day – we have early voting in Georgia – because I happened to be awake anyway. So at least in my house waking up early drives turnout.) Rain also drives down voter turnout. Perhaps if you really wanted to you could blame the results in Wisconsin on rain this Election Day… but let’s not go down that rabbit hole.

For an illustration of a similar phenomenon, take a look at the Jawbone circadian rhythm map (by Tyler Nolan and Brian Wilt), which shows that people (well, Jawbone fitness tracker owners) on the eastern side of a time zone boundary go to bed later than those on the western side of the same boundary. Interestingly, they don’t see the effect in total amount of time sleeping, which suggests that in their data set people on the eastern side of a time zone boundary also wake up later.