Here are the proper prime divisors of the first nine natural numbers (a proper prime divisor is a prime different from n that divides n evenly):

1: (none)
2: (none)
3: (none)
4: 2 × 2
5: (none)
6: 2 × 3
7: (none)
8: 2 × 2 × 2
9: 3 × 3

So, if we include repeated instances of a given factor:

• 1, 2, 3, 5, and 7 have 0 proper prime divisors
• 4, 6, and 9 have 2 proper prime divisors
• 8 has 3 proper prime divisors

Mathematicians Ana Luzón and Manuel A. Morón of Universidad Politecnica de Madrid point out a coincidence: The numerals in each of these groups have the same basic shape — within each group it’s possible to transform one numeral into another by bending, shrinking, and expanding. So, for example, it’s possible to bend a numeral 1 made of clay into a 2 or a 7, but not into a 9 — we’re not allowed to poke a new hole in the clay or to affix one part of it to another.

Luzón and Morón write that if two of these nine numerals have the same number of proper prime divisors, then those two will “cut a sheet in the same number of pieces if you write them down with a scalpel.” And if the scalpel doesn’t cut the sheet into multiple pieces, then the number you’re writing is prime (except for 1).

Note: This works only if the numeral 4 is “closed” at the top, not open. So this post will make sense if you’re reading it on Futility Closet (which uses the “closed” font Georgia), but possibly not if you’re reading it in a different font elsewhere. Maybe this tells us how 4 “ought” to be written!

(Ana Luzón and Manuel A. Morón, “4 or 4? Mathematics or Accident?” Mathematics Magazine 75:4 [October 2002], 274.)

If you apply one straight cut to a pancake, pretty clearly you’ll get 2 pieces. With two cuts, the most you can get is 4. What’s the greatest number you can produce with three cuts? If the cuts meet neatly in the center, you’ll get 6 pieces, but if you’re artfully sloppy you can make 7 (above). Charmingly, this leads us into the “lazy caterer’s sequence” — the maximum number of pieces you can produce with n straight cuts:

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, …

Generally it turns out that the maximum number for n cuts is given by the formula

each number equals 1 plus a triangular number.

A related question is the pancake flipping problem. You’re presented with a spatula and an untidy stack of pancakes of varying sizes. You can insert the spatula at any point in the stack and flip all the pancakes above it. What’s the least number of flips required to sort the pancakes in order of size? Interestingly, no one has found a general answer. It’s possible to work out the solution for relatively small stacks (in which the number of pancakes is 1, 2, 3, …):

0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, …

But no one has found a formula that will tell how many flips will get the job done for a stack of any given size.

The problem has an interesting pedigree. Bill Gates worked on it at Harvard (PDF), and David X. Cohen, who went on to write for The Simpsons and Futurama, worked on a related problem at Berkeley in which the bottom of each pancake is burnt and the sort must be completed with the burnt sides facing down.

CCNY mathematician Jacob Goodman, who first hit on the pancake flipping problem while sorting folded towels for his wife, submitted it to the American Mathematical Monthly under the name Harry Dweighter (“harried waiter”). His household chores have produced at least one other publication: After some thoughtful work with a swivel-bladed vegetable peeler, he published “On the Largest Convex Polygon Contained in a Non-convex n-gon, Or How to Peel a Potato.”

(Thanks, Urzua.)

This is charming — in his 1962 textbook Experimentation and Measurement, statistician William J. Youden ends his chapter on “Typical Collections of Measurements” with this:

Your author has a small printing press for a hobby. He set in type his opinion of the importance of the normal law of error.

In 2014, Michigan Technological University physicists Robert Nemiroff and Teresa Wilson thought up a novel way to detect time travelers: Search the Internet.

They searched for mentions of “Comet ISON,” a sun-grazing comet discovered in September 2012, and for “Pope Francis,” whose papacy began in March 2013 and who is the first of his name. Both of these subjects are historically momentous enough that they might be known even to people in the far future; if those people travel into our past, then they might mention them inadvertently in, say, 2011, before we could plausibly have done so ourselves.

“Given the current prevalence of the Internet … this search might be considered the most sensitive and comprehensive search yet for time travel from the future,” they reported, acknowledging that “technically, what was searched for here was not physical time travellers themselves, but rather informational traces left by them.”

And they note that our failure to detect travelers doesn’t mean they’re not there. “First, it may be physically impossible for time travellers to leave any lasting remnants of their stay in the past, including even non-corporeal informational remnants on the Internet. Next, it may be physically impossible for us to find such information as that would violate some yet-unknown law of physics. … Furthermore, time travellers may not want to be found, and may be good at covering their tracks.”

See Regrets and The Telltale Mart.

In 1978, inspired by this Peanuts cartoon, Nathaniel Hellerstein invented the Linus sequence, a sequence of 1s and 2s in which each new entry is chosen the prevent the longest possible pattern from emerging at the end of the line. Start with 1:

1

Now if the second digit were also a 1 then we’d have a repeating pattern. So enter a 2:

1 2

If we choose 2 for the third entry we’ll have “2 2” at the end of the line, another emerging pattern. Prevent that by choosing 1:

1 2 1

Now what? Choosing 2 would give us the disastrously tidy 1 2 1 2, so choose 1 again:

1 2 1 1

But now the 1s at the end are looking rather pleased with themselves, so choose 2:

1 2 1 1 2

And so on. The rule is to avoid the longest possible “doubled suffix,” the longest possible repeated string of digits at the end of the sequence. For example, choosing 1 at this point would give us 1 2 1 1 2 1, in which the end of the sequence (indeed, the entire sequence) is a repeated string of three digits. Choosing 2 avoids this, so we choose that.

Admittedly, this isn’t exactly the sequence that Linus was describing in the comic strip, but it opens up a world of its own with many surprising properties (PDF), as these things tend to do.

It’s possible to compile a related sequence by making note of the length of each repetition that you avoided in the Linus sequence. That’s called the Sally sequence.

Suppose k runners are running around a circular track that’s 1 unit long. All the runners start at the same point, but they run at different speeds. A runner is said to be “lonely” if he’s at least distance 1/k along the track from every other runner. The lonely runner conjecture states that each of the runners will be lonely at some point.

This is obviously true for low values of k. If there’s a single runner, then he’s lonely before he even leaves the starting line. And if there are two runners, at some point they’ll occupy diametrically opposite points on the track, at which point both will be lonely.

But whether it’s always true, for any number of runners, remains an unsolved problem in mathematics.

Write down an integer. Remove any zeros and sort the digits in increasing order. Now add this number to its reversal to produce a new number, and perform the same operations on that:

In the example above, we’ve arrived at a pattern, alternating between 12333334444 and 5566667777 but adding a 3 and a 6 (respectively) with each iteration. (So the next two sums, after their digits are sorted, will be 123333334444 and 55666667777, and so on.)

Princeton mathematician John Horton Conway calls this the RATS sequence (for “reverse, add, then sort”) and in 1989 conjectured that no matter what number you start with (in base 10), you’ll either enter the divergent pattern above or find yourself in some cycle. For example, starting with 3 gives:

3, 6, 12, 33, 66, 123, 444, 888, 1677, 3489, 12333, 44556, 111, 222, 444, …

… and now we’re in a loop — the last eight terms will just repeat forever.

Conway’s colleague at Princeton, Curt McMullen, showed that the conjecture is true for all numbers less than a hundred million, and himself conjectured that every RATS sequence in bases smaller than 10 is eventually periodic. Are they right? So far neither conjecture has been disproved.

(Richard K. Guy, “Conway’s RATS and Other Reversals,” American Mathematical Monthly 96:5 [May 1989], 425-428.)