You and your best friend Bill are the greatest bards in the kingdom— but maybe not the brightest. The hit song you performed at last night’s fair insulted the king, and now your hours are numbered. Just before your execution, you’re approached by the princess, 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 possibly 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 as well as— 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 a 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 6 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 possibly 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?
Pause here to figure it out yourself. Answer in 3
Answer in 2
Answer in 1
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 2 you’d be at 75, but without any further ladder boosts, you’d need another 5 rolls to get out. Or this one— three ladders get you to 82, but then it would take another 4 rolls to get around the pesky snake at 88. In fact, there’s no path that can get you to the exit in 5 moves using exclusively ladders. 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 backwards from those to find a 5 roll path?
There are exactly two 5-turn routes you might find using these ladders: Ladder snake snake ladder exit, and ladder ladder snake ladder exit. So that’s it, right? Unfortunately, no— 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 work, except for this snake, at exactly the midpoint between the two ladders. If Bill were to take that snake on his route, you could then get past it and up to the exit in just 5 moves. And good news— whichever path you choose, one of the 5-roll paths we found earlier does exactly 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.
