Optical illusions and puzzles share overlapping vibes. Mental trickery, ‘aha’ moments, cross-eyed frustration. Did you already know that the Neural Correlate Society holds an annual contest for the Best Illusion of the Year? This yr’s winner created a LEGO mannequin of Harry Potter’s Platform 9 ¾, full with a seemingly permeable brick wall:
This neat mirror demonstration additionally stands out amongst current finalists. This week’s puzzles have an illusory nature to them. I’ll clarify what I imply within the resolution write-up subsequent Monday.
Did you miss final week’s puzzle? Check it out right here, and discover its resolution on the backside of as we speak’s article. Be cautious to not learn too far forward if you happen to haven’t solved final week’s but!
Puzzle #19: Mental Illusions
1. 100 ants fall onto a meter stick on the identical time at random places. Each ant begins strolling towards the left or the fitting finish of the stick at random, at a velocity of 1 meter per minute. Ants proceed on their chosen course, however any time two ants collide, they each instantly reverse instructions and proceed strolling the alternative method on the identical velocity. What is the longest period of time it might take earlier than the entire ants have walked off the tip of the stick?
2. A rectangle is inscribed inside 1 / 4 of a circle that’s centered at O. Find the size of the rectangle’s diagonal AC.
I’ll be again subsequent week with the options and a brand new puzzle. Do you already know a cool puzzle that I ought to cowl right here? Message me on Twitter @JackPMurtagh or e mail me at email@example.com
Solution to Puzzle #18: The Long Hall
Did final week’s finance job interview query offer you a run to your cash?
The excellent squares (1, 4, 9, 16, 25, 36, 49, 64, 81, and 100) are the one doorways that will likely be open on the finish. Shout-out to riddler88 for deducing the explanation.
To see why, recall the definition of a divisor out of your early days in math class. The divisors of a quantity are the numbers that divide it evenly with no the rest. So for instance, the divisors of 12 are: 1, 2, 3, 4, 6, and 12. Notice that every door will get toggled throughout the rounds that correspond to its divisors (e.g. when the eighth particular person walks by, doorways 8, 16, 24, 32, and so on. are toggled, whereas door 12 isn’t touched as a result of 12 isn’t divisible by 8). Since door 12 begins closed and will get toggled a fair variety of instances (it has six divisors), it’s going to finish within the closed place. So the query turns into: which numbers have an odd variety of divisors?
Divisors have a tendency to come back in pairs. 1 multiplies with 12 to equal 12, so 1 and 12 are each divisors. 2 multiplies with 6 to equal 12 so they’re each divisors, and so forth. So the one numbers whose divisors don’t all are available pairs are numbers that may be made by multiplying a quantity with itself. For instance, the divisors of 16 are 1, 2, 4, 8, and 16. 1 pairs off with 16, 2 pairs off with 8, and 4 doesn’t have a accomplice as a result of 4 is multiplied by itself to equal 16. So the proper squares are precisely these numbers with an odd variety of divisors and people are the doorways that find yourself open.
I like this puzzle as a result of the proper squares appear to come back out of nowhere. There’s no whiff of them within the setup. Many folks have familiarity with squared numbers, however I think the characterization of them as the one numbers with an odd variety of divisors will likely be new to many readers.