Gizmodo Monday Puzzle: Can You Save the Ugly Duckling from Certain Death?

The Big Bad Wolf is going to tear the Ugly Duckling to shreds unless you put that noggin to use. Yes, I’m blending characters from two different fairy tale universes, but maybe the Brothers Grimm meets Hans Christian Andersen is the crossover event we never knew we needed.

Ensuring that the Ugly Duckling lives to grow into a beautiful swan in this week’s puzzle will be quite difficult, but there is a satisfying solution if you stick with it. Much like a fairy tale, various incarnations of the problem have existed in puzzle folklore for some time. If you already know the solution, please refrain from spoiling it in the comments. I want to give newcomers a chance at a shout-out and all of the happily-ever-after glory that comes with it.

Did you miss last week’s puzzle? Check it out here, and find its solution at the bottom of today’s article. Be careful not to read too far ahead if you haven’t solved last week’s yet!

Puzzle #28: Duck Duck Go

The Ugly Duckling is in the center of a circular lake when he spots the Big Bad Wolf lurking on the shore. The wolf cannot swim, and the duckling cannot fly away from water (but can fly from land). The wolf patrols the outside of the lake waiting for the duckling to reach land so that he can feast. The wolf moves four times faster than the duckling can swim. He can see the duckling and is free to prowl along the shoreline however he wants. How can the duckling make it to land and fly away without becoming the wolf’s dinner?

As usual, this isn’t a trick question. The answer isn’t “the lake is frozen so the duckling can fly away” or anything like that. There is a legitimate strategy with mathematical reasoning behind it. Good luck!

I’ll be back next Monday with the answer and a new puzzle. Do you know a cool puzzle that you think should be featured here? Message me on Twitter @JackPMurtagh or email me at gizmodopuzzle@gmail.com

Solution to Puzzle #27: Heads Up

Last week I asked you to analyze a twist on coin flips.

You and I are going to settle something with a coin flip. Instead of an ordinary coin flip, you’ll call either “HHT” or “THH”. Then we’ll flip the coin multiple times in a row and record the results. If the sequence heads, heads, tails occurs first then HHT wins and if the sequence tails, heads, heads occurs first, then THH wins. We keep flipping until one of them occurs.

Which do you call? Or does it not matter?

What is the probability that each wins?

THH wins 75% of the time and HHT only wins 25% of the time, so you should call THH. Shout-out to Enfy for the correct solution and for noticing that the winner is determined after only two flips.

If you flip a coin three times in isolation, the probability that the tosses turn up heads, heads, tails is the same as the probability that they turn up tails, heads, heads (both ⅛), so it’s tempting to reason that the choice doesn’t matter. But the fact that we keep flipping until one sequence occurs actually changes everything.

If the first two flips are both heads (which occurs 25% of the time), then HHT is guaranteed to win, but all other combinations of first two flips (TT, TH, and HT) lead to a win for THH (giving us the 75%). If the first two flips are HH, then we either get a tails next, giving HHT an instant win, or we delay tails by flipping more heads. Nobody wins on a string of heads, so we’ll keep flipping until a tails occurs, at which point the last three flips will be HHT.

HHT can only win if no tails precedes it in the sequence of flips. To see this, imagine a string of coin flips terminating with HHT. What coin flip immediately preceded this? If it’s tails, then THH actually won. But if it’s heads, then what preceded that? Again a tails means that THH actually won. So if the first two flips are either TT, TH, or HT, then a tails has already occurred without HHT winning, leading to a guaranteed win for THH.

This phenomenon generalizes. For any sequence of three flips that I claim as my win condition, you can find a different sequence of three that gives you an advantage over me. Furthermore, the winning sequence tends to end with the start of the losing sequence. For example, if I claim HHH, then you’d claim THH and you would win 87.5% of the time.

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