Advanced Puzzling: Math Imitates Puzzle

One thing I’ve seen that seems to be mostly loathed by lovers of escape rooms is the math puzzle, or worse, puzzle-free math computation. Though I love math, I get it- you sign up for Indiana Jones-like adventure, not arithmetic class.

Recently I encountered a problem in an escape room, though, that reminded me of how mathematics done right does scratch the same itch as a puzzle.

Take a look at the following escape room puzzle (modified slightly by me) and tell me how long it takes you.

Ready, set, go!

That’s not the same puzzle in the escape room I just did- the puzzle from the room actually had many more numbers, albeit smaller numbers, in the calculation.

You may have already figured out why I’m showing this example, but I’ll write it out anyway. The most common reaction to this “puzzle” is to start crunching numbers, writing it all out on notebook paper, and taking a few minutes for your group’s best math person to finish the numbers. But there’s a much better way.

Note the setup for the problem only requires the last digit of each answer. That alone allows us to skip the big calculation. Where it first appeared there wasn’t a puzzle, we find that a little bit of math and a big dose of common sense get us through this quickly and easily.

132432 + 327327 – 12 + 8

If we ignore everything but the ones digits:

2 + 7 – 2 + 8 = 15

So whatever else this calculation gets us, the last digit is going to be 5.

I’d be loathe to introduce this logic without providing an intuitive explanation of why it’s the case, so I’ll provide a little summary.

Any time you see a number with more than one digit, you can isolate the ones digit easily. 132432, for example, can be substituted for 132430 + 2. Doing that for the full calculation here gets:

(132430 + 2) + (327320 + 7) – (10 + 2) + 8

The commutative property of addition means we can reshuffle these to get:

(132430 + 327320 – 10) + (2 + 7 – 2 + 8)

Even though calculating that set on the left would be a minor pain, we don’t have to do it. With the ones digit being zero in all three cases, we know that will yield a big number ending in zero, and that our ones digit will come from the simple calculation on the right.

(some big number that ends in 0) + (15)

Our last digit will be 5 in the answer.

Not only is this a reminder to always keep an eye out for a better way to do things, but it’s also a call to watch for cases where life imitates puzzle. The way we look at things often makes a huge difference in whether it seems impossible or easy.

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