## Depressing Expressions

*January 26, 2012 at 11:18 am* *
7 comments *

Did you hear that Monday, January 23, 2012, was the most depressing day of the year? That’s according to Cliff Arnall, a British life coach who, for a little while, was a tutor at Cardiff University. He used the following formula to make his prediction:

In that expression, *W* = weather, *D* = debt, *d* = days until next payday, *T* = time since Christmas, *Q* = time since a failed quit attempt (such as abandoning a New Year’s resolution), *M* = motivation level, and *N _{a}* = need to take action.

When I heard that a psychologist was creating mathematical expressions, I had just one thought:

Why did the psychologist send the expression to a doctor?

Because he wasn’t being rational.

When I read the formula, my first thought was, “Wow, that’s an incredible bunch of rubbish!” (Funny, I don’t normally use the word *rubbish*. Maybe it happened because I read about the expression in a British newspaper?) Only *W* and *T* are universally measurable variables. While *D*, *d*, and *Q* are also measurable, they vary from person to person and shouldn’t be used to predict a global most depressing day. And what’s this nonsense about time from Christmas? Is that really a factor for Jews, Sikhs, and other non-Christians? (Note: Many online sources incorrectly state that *d* = monthly salary. But that would cause the formula to make even less sense.)

This expression has been used for several years to predict the most depressing day, and a similar expression has been used to predict the happiest day of the year. The happiest day expression, which is similarly unintelligible, regularly predicts a date in June. Interestingly, the “research” was sponsored by Wall’s Ice Cream. (Hmm, now why would an ice cream company have an interest in people being happy during the summer?)

It’s a little too early in the year to say that January 23 will be the most depressing day of 2012. In fact, January 23 isn’t even the most depressing day *so far* in January — that distinction belongs to Sunday, January 8.

As for crazy expressions, the following equation contains my favorite:

The value of *k* doesn’t matter, but the equation doesn’t hold if the placeholder variable *k* is not included.

Incidentally, this equation is related to the following problem: Raise *n* + 1 consecutive integers to the power *n*. Subtract the first from the second, the second from the third, and so on, until you’re left with a set of *n* integers. Then subtract the first from the second, the second from the third, and so on, until you’re left with a set of *n* ‑ 1 integers. Continue this process until you’re left with just one integer. Its value may surprise you.

Entry filed under: Uncategorized. Tags: Cliff Arnall, depressing, equation, expression, happiest, integer.

## 7 Comments Add your own

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1.Brent | January 26, 2012 at 8:57 pmWoah, that’s a neat identity! But surely it still holds if k is not included: this is equivalent to setting k=0.

Also, in the last paragraph I think you meant to say “raise (n+1) consecutive integers to the power n”?

2.venneblock | January 27, 2012 at 8:17 amArrgh! Why are things so clear in my mind so hard to transfer to a blog post? I swear I had it correct in my handwritten notes. Thanks for pointing it out.

And good point about

k. I noted that its value didn’t matter… but I didn’t think to send it to zero. I guess that’s why you were a MathCounts superstar, while I was relegated to water boy for the table tennis team.3.Brent | January 27, 2012 at 8:25 amHah! If it makes you feel any better, I still haven’t been able to prove the identity after messing around with it for an hour…

4.Brent | January 27, 2012 at 8:10 pmWell, I have a neat combinatorial proof now. =) I’ll write about it on my blog soon…

5.Differences of powers of consecutive integers | The Math Less Traveled | January 29, 2012 at 4:17 pm[…] Vennebush of Math Jokes 4 Mathy Folks recently wrote about the following procedure that yields surprising results. Choose some positive integer . Now, […]

6.Travels in a Mathematical World | April 5, 2012 at 4:40 amCarnival of Mathematics 85……a series of four posts in response to Depressing Expressions by Patrick Vennebush over……

7.A combinatorial proof: reboot! | The Math Less Traveled | August 31, 2019 at 7:18 am[…] than seven years ago I wrote about a curious phenomenon, which I found out about from Patrick Vennebush: if you start with a sequence of consecutive th powers, and repeatedly take pairwise differences, […]