Here's a fun clip from BBC Two's collegiate quiz show University Challenge. Host Jeremy Paxman asks the contestants what day of the week it will be 100 days after Monday. A player named Binnie responds quickly: "Wednesday." Paxman's face contorts. "How did you know that?" he asks, astonished.
"Modular arithmetic," Binnie replies, matter-of-factly.
What's wonderful about this exchange is that, contrary to Paxman's sarcastic quip, modular arithmetic – at least, to the extent that it applies to this quiz show question – is pretty straightforward.
Without getting into modulo terminology (for that, see see the end of this post): There are seven days in a week, and seven goes into 100 days fourteen times, with two days left over. Monday plus any-integer-multiple-of-seven days will bring you right back to Monday, so this remainder of 2 days is the number Binnie "adds" to Monday to get his answer of Wednesday.
Written out like this, the method for solving this problem couldn't be more obvious. So why is Paxman so astonished at Binnie's speedy reply?
If I had to guess, I'd venture that fancier forms of calendar calculation (Kim Peek, for instance, was renown for being able to tell you which day of the week you were born) have conditioned us to treat questions like this as impenetrable, and the ability to answer them quickly as indicative of savant-levels of intelligence – when, in fact, if your times tables are halfway decent, the mode of attack is actually pretty straightforward.
I'm not saying Binnie isn't smart, or that his response-time isn't impressive (he could be a bonafide genius for all I know, and I'll be the first to admit that the solution to this problem was not immediately obvious to me). I'm saying there's a lesson to be learned here about how we approach daunting questions, whether they're scientific conundrums, difficult tasks, or game show queries. What I'm saying is the difference between an unsolvable problem and a tractable one can occasionally be traced to our first impression of the challenge at hand. ( Preparedness, of course, never hurts. This is one of the reasons puzzles and brain teasers, which condition our minds to identify familiar approaches to seemingly unfamiliar problems, are such valuable forms of mental exercise.)
*The modular arithmetic Binnie uses is "arithmetic modulo 7." The steps he takes can be described by the phrase "100 modulo 7," which can be shortened to "100 mod 7," which basically translates to: "the remainder you get when you divide 100 by 7."
Here's more, from Khan Academy's "Intro to Modular Math":
When we divide two integers we will have an equation that looks like the following:
A / B = Q with remainder R
A is the dividend
B is the divisor
Q is the quotient
R is the remainder
Sometimes, we are only interested in what the remainder will be when we divide A by B.
For these cases there is an operator called the modulo operator (abbreviated as mod).
Using the same A,B,Q,and R as above, we would have: A mod B = R
We would say this as "A modulo B is congruent to R". Where B is referred to as the modulus.
For example we know: 13/5 = 2 remainder 3
13 mod 5 = 3
In the case of the video above, 100 mod 7 = 2. Monday plus 2 days = Wednesday.
Another handy way to think about modular arithmetic is with with a 12 hour clock. If it's currently 8:00, seven hours later it will be 3:00 (rather than 15:00), because 15 mod 12 = 3.