The Cheating Death Riddle

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You and your best friend Bill are the greatest bards in the kingdom— but maybe not the brightest. Your hit song has insulted the king and now you’re slated for execution. Luckily, Death is a connoisseur of most excellent music and has decided to give you a chance to escape your fate. Can you beat him at a life-sized game of Snakes and Ladders and live to sing another day? Shravan S K shows how.


You and your best friend Bill are the greatest bards in the kingdom— but maybe not the brightest. The hit song you performed last night’s fair insulted the king, and now your hours are numbered. The princess approaches you before your execution, who happens to be a big fan. She slips you a pouch containing two magic, six-sided dice that can land any way you wish. You have no idea how this could help, but you thank her anyways.

As everything goes dark, you both suddenly find yourself standing in a deep cavern before a robed figure. It turns out that Death is a connoisseur of most excellent music. He’s decided to give you a chance to escape your fate by beating him at his favorite game: life-sized Snakes and Ladders. The cavern has 100 rooms connected by passageways and— you guessed it— snakes and ladders. Each turn, one of you will roll a die, then advance that many rooms. Ending your move at the bottom of the ladder means moving up to the top while ending on a snake’s head will make you slide down to its tail. Each of you must make your way through the cavern and reach the exit in fewer turns than Death himself.

Before you can think twice, your host starts casting his die, and within six rolls, he’s reached the cavern entrance. “Better get moving,” he says. “Oh, and by the way, once one of you uses a snake or ladder, it’ll disappear and be inaccessible to the other.” You have no idea how you can get lucky enough to win, but then you remember your magic dice. Can you figure out a sequence of rolls that will get both of you to the exit in 5 or fewer moves each?

The cavern contains many routes to the exit, and it’s natural to gravitate to the ladders that allow you to cover a lot of ground quickly. Like this one— on turn two, you’d be at 75, but without any further ladder boosts, you’d need another five rolls to get out. Or this one— three ladders get you to 82, but then it would take another four rolls to get around the pesky snake at 88. No path can get you to the exit in 5 moves using ladders exclusively. So, where does that leave you? Snakes. It’s counterintuitive to make moves that take you farther away from your target, but snakes open up a surprising amount of opportunities. For instance, these three ladders would put you a single roll from the exit but cost too many turns to reach by ladders. These two are, however, only 5 and 12 rooms from snake tails, respectively. So, could you work backward from those to find a five-roll path?

You might find precisely two 5-turn routes using these ladders: Ladder snake ladder exit and ladder exit. So that’s it. Unfortunately, the two paths share a snake and a ladder, each of which will be unusable after the first of you slides and then climbs. You won’t find 5-turn moves anywhere else, so is one of you doomed?

There’s one more possibility to consider: the disappearing ladders and snakes can impede your progress, but could they also help it? There are two paths to the 94 ladder that almost works, except for this snake, at exactly the midpoint between the two ladders. If Bill were to take that snake on his route, you could get past it and up to the exit in just five moves. And good news— whichever path you choose, one of the 5-roll paths we found earlier does precisely that without using any of your snakes or ladders. So all you need is for Bill to go first and stick to the plan.

You and Bill both reach the cavern entrance and climb back into the public square, to everyone’s surprise. Before the guards can seize you again, you wow the crowd— and king— with a song about your epic contest with Death.

Ali Kaya


Ali Kaya

This is Ali. Bespectacled and mustachioed father, math blogger, and soccer player. I also do consult for global math and science startups.

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