Last night at around 9 pm my mom called.
(Hey! I just realised I didn’t even tell her about the Captain’s report card! Doh!)
Anyway, she was working on a math problem with Red, my 14-year-old niece. Here’s what they do in grade 9 math these days.
There’s a drawing of a hexagon with a circle at each veritice, and another circle at the midpoint of each edge, like so:
a b c
l d
k e
j f
i h g
Arrange the first 12 counting numbers in the 12 circles so that the sum of the three numbers of each of the six sides is the same. The three circles connected by the triangle connecting the three vertices (a-e-i) also give the same sum.
My mom was calling to find out if “the first 12 counting numbers” meant the numbers 0-11, or the numbers 1-12. It’s a term I am not familiar with at all. I guessed it meant 1-12, because if you ask the Captain to do some counting, he starts at 1, not 0. Turns out, though, [SPOILER!] the answer is to use the numbers 0-11.
and I are always into doing a little math in our free time, so we set up a bunch of algebraic equations and went about solving them. But you know what? There doesn’t seem to be any way to actually solve the problem.
The best we could do is find boundaries on S, where S is the sum of each of the sides (and the triangle in the middle). We could set a minimum value of S (14) and a maximum value of S (19), and as it turns out, you can make the hexagon alone work with any number in that set (i.e. you can make a hexagon where the sum of each side is 14, 15, 16, 17, 18 or 19). And in fact, for most of those possible sums there are multiple solutions for the placement of the numbers around the hexagon. But we believe that the triangle-sum limitation means that only one solution is actually possible…but we could not solve it using algebra.
In the end we had to look it up on the internet and found this page that talks about finding the range of possible values for S. By fluke the chosen example they had for the hexagon shape is the solution that Red needs. But even this page, which uses equations and approaches the problem mathematically, can only get so far before it’s just guesswork.
was up until midnight working on this, only to give up in frustration. He says that they may as well have sent Red home with a Sudoku, for all the math she is actually learning by working on this problem. Certainly the algebraic approaches we used to even limit the value of S were, we believe, way beyond what they do in Grade 9 math, so probably she was expected to figure this out purely through trial and error. Surely there are better things to be teaching in math these days!
TurtleHead 
Good lord! That’s impressive.
Whether you use 0-11 or 1-12 is irrelevant, right? If you have a solution using 0-11, and then add one to every number, you get a solution for 1-12 (where the common sum is 3 larger than in the other solution).
Slightly tangentially, I have this long-running internal debate about whether I should teach our children to count starting at 0. As a mathematician, I feel pretty strongly about 0 (extremely strongly, in fact). But I don’t want my kids to be outcasts, or get confused because everyone else in his class is fixated on 1.
Huh. You are correct about the 1-12 versus 0-11 thing. Man, I feel embarrassed!
This being true, I feel like the teacher probably wanted them to use 1-12. I called my niece this morning to give her the 0-11 solution. Maybe I should call her back to correct it.
Although, re-reading your comment, I get the impression that you consider “0” to be the first “counting” number anyway. Is this a generally accepted mathematical thing? Just wondering.
In high school I was taught to distinguish between the set N of “Natural numbers” (which didn’t include 0) and the set W of “Whole numbers” (which did). However, mathematicians rarely use the term “whole numbers”, and the debate (about which there is certainly no agreement) is whether you consider 0 to be a member of the natural numbers (which I’ll refer to as N).
From a mathematical/logical/aesthetic point of view, I think including 0 in N is absolutely the right choice. There’s a beautiful
construction of N from basic set-theoretic axioms due to von Neumann, which to me is the most “natural” way to describe the set N, and that construction rather “naturally” includes 0. But of course the natural numbers arose quite “naturally” in the context of counting, for which it’s less obvious that 0 is appropriate (though even there I believe it could be justified; it’s just that it’s not obvious that it belongs there at the outset).
Interestingly, Mr Excitement has a counting puzzle which consists of the numbers 0 through 9. I think that’s great. I think I don’t want to be a hardass about 0 though; I’ll just try to introduce him to it early. But one thing I can’t stand is when i see a toy where 0 is used but it follows 9. That just makes no sense at all.