## Math Clock Showdown

When shopping for gifts for someone, there are a few wells from which one frequently draws inspiration. A person's favorite TV show, for example, or favorite band; such preferences can often provide good fodder for gift ideas. One's career can also be included in this list - in my case, the result is that I am frequently the recipient of math-themed paraphernalia.

I've written before about my mixed feelings regarding math t-shirts. Today, though, I'd like to tackle a different type of gift: the math clock. This is inspired, in part, by a gift I received from my grandmother (bless her heart) over the holiday. The gift, pictured below, was an analog clock in which the numbers have been replaced by (what one would hope to be) mathematically equivalent expressions.

Figure 1: Clock with a black background.

Don't tell her, but we haven't yet put this clock up in our apartment. In my own defense, this is mostly because we already have a math clock displayed prominently in the kitchen. My future wife says that she gave me our first clock, and this is a claim I have no reason to doubt.

Figure 2: Clock with a white background.

Sadly, our apartment is simply not big enough for two mathematically themed clocks. The question becomes, then, which one should be kept in-house, and which one should be relegated to the dungeon of an office I hold at the math department? It seems that the most natural way to answer this question is to compare the mathematics of each clock side by side.

Let's start at the top and work our way around.

**12: **6 · 2 vs.

The black clock has a simpler expression, but perhaps it's a little too simple. At least the white clock asks you to work a little for it. Point: white clock.

**1: **102,413 – 102,412 vs.

The white clock's expression for 1 is a little to esoteric for my taste. The notation is meant to symbolize Legendre's constant - this number is related to the asymptotic behavior of prime numbers, and historically it was believed to be greater than 1 for some time, based on numerical evidence. But if you didn't know all that, there's no way you could make the connection between the notation and the number 1. At least the expression in the black clock makes the connection to 1 clear. With a little more information so that the casual time-teller could have learned something about primes, the white clock may have had the upper hand. As it stands, though, I must side with the black clock here.

**2:**

Here I must tip my hat to the infinite sum. The square root is nice, but the sum is nicer, and if you're trying to impress a date who doesn't know about geometric series, this will provide you with an excellent opening. Point: white clock.

**3: **198 ÷ 66 vs. some XML garbage.

Come on, white clock. That isn't even math! Point: black clock.

**4:** 50/2 = 100/*x* vs. 2^{-1} mod 7

I prefer the poetry of including "clock arithmetic" on the face of a clock. Plus, modular arithmetic (as it is more professionally known) is a topic that the general population is not always exposed to, even though it's not hard to explain. I'll take any opportunity I can get for a clock to educate the masses. Point: white clock.

**5:** 630 ÷ 126 vs. (2 φ - 1)^{2}

Given my stance on 1, this may seem a little hypocritical, but I'm going to give the edge to the white clock here. Part of the reason is that the black clock loses steam pretty quickly - out of the 12 numbers, 3 are expressed in terms of long division. Come on, guys.

Besides, since we know that the expression on the white clock equals 5, this allows us to solve for φ and obtain the golden ratio. This is something someone could discover on his or her own, perhaps with the aid of something like Wolfram Alpha. So the comparison to the 1 o'clock entry isn't quite apples to apples. Or at least, that's what I'll keep telling myself.

**6:** vs. 3!

The factorial is a little less conventional, but every student should encounter it at some point. Here I'm giving the edge to the white clock again.

**7:** 52 – *x*^{2} + *x* = 10 vs.

Presumably, the black clock wants us to solve for *x*, using the quadratic formula or something. I don't get it, though - if they're going to express 7 as an unknown in a quadratic function of *x*, why would they write an equation that has two solutions, one of which isn't 7? Since 52 – *x*^{2} + *x* = 10 is the same as 42 – *x*^{2} + *x* = 0, here's a graph of 42 – *x*^{2} + *x*, so you can see the two roots:

I realize I'm being a little pedantic (after all, there isn't any negative 6 o'clock), but it would've been just as easy to write a quadratic that had only 7 as its root. Here's one: *x*^{2} – 14*x* + 50 = 1.

Besides, the white clock's entry for 7 is good in its own right. No contest here, white gets the point.

**8:** vs. some dots

I'll give it to the black clock here. The white clock is expressing 8 in base 2, but I don't know why they don't do it using digits. Probably because they also play around with the base in the next number, where they write 9 in base 4 as 21_{4}.

**9:** 3(π – .14) vs 21_{4}

If I could give negative points, I would give them to the black clock here. Their expression doesn't evaluate to 9; instead, the clock only perpetuates common misunderstandings about the number $latex \pi$. Admittedly, the black clock does give a fairly good approximation, but I've never heard of 9.004778… o'clock.

**10:** -8 = 2 – *x* vs.

These are both worthy contenders. For the sake of fairness, since I gave the white clock the point for 3!, I'll choose the black clock here in favor of the white clock's binomial coefficient.

**11: **1221 ÷ 111 vs. some hexadecimal representation of 11.

The long division is redundant, but in a sense, so is the white clock's entry - we've already seen two other cases of representing a number in a different base. In this case, I'll defer to the one that's clearer. Point: black clock.

By my count, the final score is 7 points for the white clock, 5 for the black. It was a close match, but it looks like a decision has been made. Regardless of the outcome, though, both clocks have their share of problems.

I should point out that, somewhat surprisingly, these are not the only math clocks on the market. Here are even more examples. The one which speaks to me the most, though, is probably the last one.

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