Guess the Graph on Derangement Day

Today is 4.1.23, a date in which the digit 4 occupies the first position, the digit 1 occupies the second position, the digit 2 is in the third position, and the digit 3 is in the fourth position. That is, no digit n is in the nth position. In mathematics, we call this a derangement, a permutation of the elements of a set such that no element appears in its original position. Using the same format, when will the next “derangement day” occur?

Today also happens to be April Fools Day. If pranks are your thing, then here’s a math prank for you:

What is the product of all single-digit prime factors of 143?

Why is it a prank? Because the prime factors of 143 are 11 and 13, neither of which is single‑digit. So, the set of all single‑digit prime factors of 143 is the empty set { }, which means it contains no elements. Since there are no elements in the empty set, the product of the elements should be 0, right? Well, not exactly. By convention, the product of the elements of the empty set is 1, though it might be more precise to say that the empty product is the multiplicative identity.

But my apologies for that discursion. Let’s return to the reason for today’s post.

Today is also the day when the two semifinal games of the NCAA Men’s Basketball Tournament will be played. Three teams that have never made it this far before — Florida Atlantic, San Diego State, and Miami (FL) — join UConn in the Final Four. And that leads to the math question of the day: identifying the data set that was used to construct the bar graph below.

click to enlarge the image for a better view of the axis values

The omission of axis labels and chart title was deliberate. What should they be?

That’s a big question. Maybe too big. Let’s start with two smaller questions.

What do you notice?

• Did you notice that the bars are clustered together?
• Did you notice that there are gaps?
• Did you notice that 0 occurred most frequently, and that the bars generally decrease in size from left to right?
• What else did you notice?

What do you wonder?

• Do you wonder if the graph was horizontally truncated, and maybe there are some bars with category values greater than x = 60? (Ooh, good question! There aren’t; none of the data had a value greater than 55.)
• Do you wonder why the bars only occur when the units digit is 0‑5? Or why no bars occur when the units digit is 6‑9?
• Do you wonder why I spend so much time asking annoying, rhetorical questions? (Yeah, me, too.)
• What else do you wonder?

Take a minute to analyze the graph and think about those questions. What are your initial thoughts for the data set that was used to create it? Make a first guess at what the axis labels and chart title might be.

Okay, now some hints. I mentioned above that today is the Saturday of the Final Four, so perhaps you already thought that the graph might have something to do with college basketball. If so, you’d be right.

Not enough? Then let me divulge that the data set used to create this graph contains 320 elements.

Still not enough? Don’t sweat it. This is a tough one. Your final hint is that 320 = 64 × 5.

Okay, time to let the cat out of the bag. March Madness includes 64 teams (if you don’t count the play‑in games, and — because I’m a purist, or because I’m old — I don’t). Further, every coach identifies 5 starters for each game. The 320 elements in the data set are the uniform numbers for the 5 opening-round starters from each of the 64 teams. In college, basketball uniform numbers are not allowed to include the digits 6, 7, 8, or 9. Specifically, under Rule 1, Section 22, Article 7, Clause b.2 of the NCAA Men’s Basketball Rules Book, “The following numbers are legal: 0, 1, 2, 3, 4, 5, 00, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, and 55. Team rosters can include 0 or 00, but not both.” Why does this rule exist? To help the person keeping score. No, seriously. When a player commits a foul, the referees signal the player’s uniform number to the scorer’s table using their fingers. If a player wearing 51 committed a foul, the ref might display this:

So, you think you’re the perfect college basketball player? Sorry, you can’t wear 6 or 28. You think you’re a prime-time player? Too bad, you can’t wear 7, 17, 19, 29, 37, 47, 59, 67, 79, 89, or 97. Is your last name Kaprekar? You’re outta luck, you can’t wear 9 or 99. (I know, that’s a deep cut. Look it up.)

[Update, 4/5/23] On 4/1/23, when I asked when the next derangement day would occur, it was nothing more than an innocent question. Honestly, I hadn’t even considered what the answer might be, and I thought maybe one of the readers would posit a suggestion. But as I lay in bed this morning — thinking about the math of the date, as I’m wont to do — it occurred to me that today could be considered a derangement day.

Today’s date is 4/5/23, and the digits form the set {2, 3, 4, 5}. None of the digits in the date occupy the same location in the set. For instance, the digit 4 is the first digit in the date, but it’s the third element in the set. Consequently, this might be considered a derangement day. By that logic, tomorrow, Friday, Saturday, and Sunday this week would also be derangement days.

But I don’t love that. I feel like a derangement day should contain the digits 1‑n. But I never stated that. Perhaps all we really need is a better definition of derangement day, so here it is:

A derangement day is a calendar date that contains distinct digits 1‑n such that the digit k never occurs in position k of the date.

But even in that case, the next derangement day isn’t so far away. It occurs in just 10 days: 4/15/23. In that date, the digit n never occurs in the nth position.