### The old perilous-bridge-at-night-with-dying-flashlight puzzle

#### by malmskog

A new puzzle! A family of four is on one side of a rickety bridge across a deep and otherwise uncrossable chasm. They desperately need to get to the other side. Perhaps they are being pursued by monsters. I don’t know. But I do know that it’s night time–totally dark, no moon. The bridge is a weak thing, which can only support the weight of two people at any given time. And it doesn’t have a hand rail, or anything like that, so you could walk right off the edge if you can’t see where you’re going. So anyone who is on the bridge had better have a flashlight with them. Our family of four has a flashlight. Just one, though, and the batteries are dying. In fact, there are only 17 minutes of battery power left. The family knows that it will take one of them (call her Grandma) 10 minutes to cross the bridge. Another of them (Mom) will need 5 minutes, another (fast kid) will take 2 minutes, and another (really fast kid) will take 1 minute to cross. If two people are on the bridge together, they will have to travel at the rate of the slower person, because they have to share the flashlight. There is no funny stuff, like throwing the flashlight (or throwing Grandma) across. The question, obviously, is how can the whole family get across in 17 minutes or less? Can you save this family?

Figure it out! Send solutions or questions to mathmostly@gmail.com by Tuesday March 6. Best solutions will receive an incredible math mostly/somewhat science T-shirt! Listen to Somewhat Science from 2:30-3:00 Friday afternoon (Eastern time) on WESU 88.1 FM (wesufm.org) for the solution. And a lot of other great science related stories from Wesleyan students. Speaking of Wesleyan students, big thanks to Jed Rendleman, my Math Mostly co-host this week.

Solution to last week’s puzzle: Congratulations to James Ricci and Katherine Mullins, the two winners of last week’s Math Mostly write-in puzzle explosion! The puzzle was called the Trivium’s Checkerboard, but is commonly (and slightly gruesomely) known as the mutilated chessboard problem. Their solutions are so good that I’ll just quote ’em.

James: “Milo noticed that the checkerboard alternates between black and white squares. That is, adjacent to any black square there are only white squares, and vice versa. Therefore each individual domino must cover exactly one white and one black square at once. However, there are exactly 32 black squares and 30 white squares! So no matter how he places the dominos down he will only ever be able to cover at most a set of 30 pairs of black and white squares leaving at least 2 black squares uncovered.”

Katherine: “White squares are only adjacent to black squares on Trivium’s board, and vice versa. Thus, any domino that covers two adjacent squares must cover one white square and one black square. Two white squares were removed from the checkerboard, so Trivium’s board has 2 more black squares than it has white squares. If the only dominos available necessarily cover one white square and one black square, then at least 2 black squares remain uncovered at all times.”