GeekDad Puzzle of the Week: Math Pirates - Solution

Avast! Math pirates make no apologies for this week's puzzle ye scurvy dogs! The newest pirate aboard the Euclidean Blood be "dorkusmonkey," and a fine pirate name that be. Dorkusmonkey gets t' plunder $50 of ThinkGeek booty! To all the rest of y' grog-snarfing swabs, you can read the solution and pillage ThinkGeek for $10 off'n yer next buy of $30 or more after the jump. Ar.

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Two pirates of the Euclidean Blood find themselves in port at
Pasty Pete's Inn. After a long night of carousing and other unmentionable acts, they head to their cabins with others in the
Barkadeer suite of the inn.

PEGLEG JACK FUNCTION: Yarrr. Like you and I, Pi me bucko, those three lubbers there all have a cabin here in the Barkadeer. Look sharp, ye parrot-loving sea bass! The numbers on the door of their cabins add up to twice the number of yours. If'n ye multiply them together their three numbers make 1,260. The numbers of our five cabins are, of course, all different. Yarr.

EIGHT-TOES PI: Aye, Jack. But that no tells me what their numbers be.

PEGLEG JACK FUNCTION: That be true, Pi. That be true. But head to the fore and it will if'n I tells ye that the number of me own cabin be greater than that of any of your'n. Arr.

The two pirates see that the Barkadeer suite has cabins numbered from 2 to 222.

Find the numbers of the cabins of Pegleg Jack Function, Eight-Toes Pi, and the three land lubbers they are talking about.

Note from Capt. Calico Hexahedron: To all you jelly-legged, squid-skinned, land lubbers, Aye, there be many possibilities, but
Pegleg Jack says that ye can choose if'n ye know what he says, which means Pi must narrow his choices. Yar.

SOLUTION

Immediately, you will see that the possibilities of four numbers will turn your head to chum. However, because we are told by Jack that Pi will be certain of the numbers once Pi is told Jack's number is higher, Pi knows that all possibilities but one can be eliminated. The only way to do that is narrow your choices to two sets of three different numbers that are factors of 1,260 with a common even-numbered sum (since the sum is twice Pi's number).

The two sets of cabin numbers that fulfill this are: 4, 9, 35 and 5, 7, 36 (each with a sum of 48). You can assume Pi knows Jack's cabin number, or not, but the fact that he can choose with certainty if he knows Jack's cabin number is higher limits the choice to only one set and eliminates the other, therefore Pegleg Jack Function's cabin is number 36 (if it were higher, both sets would be true and would not allow Pi to choose with certainty).

Pi retires and airs out his eight toes in cabin 24, whilst Jack hangs his peg leg in cabin 36. The three lubbers go to cabins 4, 9, and 35.

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