Tuesday, March 24, 2015

Sherwood Forest

"A three-way duel is an old tradition in these lands," said Robin Hood, "because it is unlikely that between two champions, there shouldn't come a third party in between, to show he is the bravest.

Miller, Allan-a-Dale, and Brother Tuck now battle like real thieves, and may the best win. Here are the crossbows, each of them with its own set of arrows. The three crossbows are all slightly different, and each set of arrows works only on the crossbow it was designed for. Unfortunately, only one set of arrows is flawless: a quarter of the set of arrows of the second crossbow is faulty, as is an incredible half of the third set. A faulty arrow is indistinguishable from a good one until it is shot. Now, go on, pick a straw, and let fate be fulfilled."

Robin Hood showed his clenched fist, from which three straws stuck out. Miller was the luckiest, having picked the longest straw, while Brother Tuck stared at his short straw with an expression of dread, for he knew full well that none of the three men would miss a target with a good arrow. "Well, Miller, here it is: the king of all crossbows. You have a 100% chance of killing your target. Allan-a-Dale, I hand you the crossbow with a few faulty arrows: with it, your chances are 75%. And there you are, Brother Tuck: the 50% crossbow. But then, a religious man like you could even duel with only your spiritual belief."

The fit monk breathed deep, and muttered, "God almighty, Robin, you want to see me dead. It is not possible to duel in these conditions."

"You are right, Brother Tuck," replied Robin Hood, after thinking for a little while. "Here is what we will do. You, Brother Tuck, will have the right to shoot first, after choosing your desired position on the field. Friends, please remember the rules: you will take position on the corners of an imaginary triangle, each of you 80 yards away from the other two. The order in which you will take turns to shoot is clockwise. On your turn, you may choose which of the other two you wish to shoot. Get ready: it is time to duel!"

Brother Tuck gulped as he chose his position, a taunting raven fluttering overhead squawking songs of doom.
sherwood forest
What position (A or B) did Brother Tuck choose, and which adversary did he shoot first, in order to have the highest chance of survival?

Sherwood Forest Puzzle Solution

Since the order of turns was clockwise, Brother Tuck went to position B, and deliberately missed his firts shot (perhaps he should shoot the raven).

Assuming that missing the first shot gives Brother Tuck the best chance that one of the dangerous adversaries gets killed from the other before becoming himself the target (obviously, both Miller and Allan-a-Dale opt to use their first turn to shoot at the most dangerous adversary), it is necessary to consider the chances of shooting from position A and the chances from position B.

By assuming position at A, and missing the first shot (as it is right in the circumstances), Brother Tuck knows that he wouldn't be Miller's target, because Allan-a-Dale is more dangerous. Allan-a-Dale would certainly be killed (Miller's crossbow has 100% chances of hitting the target). At this point, it would again be the monk's turn (with Allan not being in a condition to take his turn), who has a 50% chance of hitting the other survivor, ie Miller, and to therefore end the duel. Obviously, if the religious man misses, then he could only get on his knees and pray, because Miller's next shot would be faultless. So, by choosing position A, Brother Tuck has a 50% chance of survival.

If Brother Tuck chooses position B, and deliberately misses his first shot, the next turn would be Allan-a-Dale's, who would choose the most dangerous adversary, ie Miller. At this point, two different outcomes can happen:

1. Allan misses Miller (25% chances); at this point, Miller will use his turn to shoot Allan. Then it will be Tuck's turn, who will have one chance out of two to pick one of the good arrows (so, it's 50% chances, but it's 50% of 25%, ie 1/8).

2. Allan hits Miller (75% chances); at this point, Brother Tuck will shoot Allan, with 50% chances of success; but this is 50% of 75%, ie 3/8 which, if added to 1/8 of the first outcome, gives Tuck 4/8, ie 50% chances of survival. In case that his shot misses, Tuck would not be a sure victim of Allan's next shot, who might miss with 25% chances. This gives the monk a further 3/32 probabilities (ie 3/4 * 1/2 * 1/4).

The calculation, as shown here, could keep going on and on until the last arrow, and demonstrates that, by choosing position B, Brother Tuck has, besides the 50% chances also offered by position A, a long string of small chances (3/32, 3/256, etc), that are possible if Allan-a-Dale misses at least one (his first, and eventually the next ones) shot at the monk.