You and yours, sinking. Here's how to survive. Image: Flickr/ChrisFriese cc license.

Kimberly Johnson said, “Never ruin an apology with an excuse.” I’m not quite sure who Kimberly Johnson is or was, but she’s included at QuoteGarden, so she must be wise beyond reckoning and I’ll take her advice: I’m sorry I’m late on the answer to this week’s puzzle.

You’re on an Italian cruise liner piloted by a cowardly megalomaniac when it slams into a reef and goes down, trapping you below the waterline with 99 other passengers. Each person uses 23 liters of pure oxygen per hour and the room starts with 250,000 liters of air at 20% oxygen. Each hour, you have a 5% chance of being rescued. Each person who tries to swim for safety has a 1% chance of reaching the surface and alerting help, thus immediately saving the remaining trapped passengers. When the room runs out of oxygen, everyone remaining dies instantly.

What strategy offers the best chance of the most survivors?

This is actually a tricky, tricky problem (and like many mathematics and statistics problems that hit the real world, it requires some assumptions — which we could probably argue about until the proverbial cows reach whatever place they call home, but let’s not). Huge geek props to our handful of correct entrants this week including Patrick, Zach, and Andy — you should send two swimmers immediately, resulting in an expected 66.94 survivors.

Mike also sent this compelling answer: “Maximum survival odds occur by allowing all 100 passengers to wait until the oxygen is nearly depleted (21.7 hours – ε), then sending swimmers out sequentially until one of them reaches help. Waiting for help produces an expected recovery probability of 67% over the 21.7 hour period. Sending out swimmers boosts the probability of at least one survivor to 88%. The expected number of survivors is 79.42.”

I would love to see some Shaolin throwdown between these two mathematically gifted factions. Any takers? Did Mike somehow gob the expected recovery probability over 21.7 hours? Is Mike looking at a smooth curve across time whereas Patrick, Zach and Andy are looking in ratcheting increments?

In any case, drawn from these three correct entrants and one rather compelling non-correct-ish entrant, the winner of this week’s puzzle (or, last week’s at this point…) and the \$50 ThinkGeek gift certificate is Patrick! Congrats!

The rest of us can use the code GEEKDAD37MD to get \$10 off a \$50 order at ThinkGeek.