This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

Two fifty foot ropes are suspended from a forty foot ceiling, about

twenty feet apart. Armed with only a knife, how much of the rope can

you steal?

logic/ropes.s

Almost all of it. Tie the ropes together. Climb up one of them. Tie

a loop in it as close as possible to the ceiling. Cut it below the

loop. Run the rope through the loop and tie it to your waist. Climb

the other rope (this may involve some swinging action). Pull the rope

going through the loop tight and cut the other rope as close as

possible to the ceiling. You will swing down on the rope through the

loop. Lower yourself to the ground by letting out rope. Pull the

rope through the loop. You will have nearly all the rope.

logic/same.street.p

Sally and Sue have a strong desire to date Sam. They all live on the

same street yet neither Sally or Sue know where Sam lives. The houses

on this street are numbered 1 to 99.

Sally asks Sam "Is your house number a perfect square?". He answers.

Then Sally asks "Is is greater than 50?". He answers again.

Sally thinks she now knows the address of Sam's house and decides to

visit.

When she gets there, she finds out she is wrong. This is not

surprising, considering Sam answered only the second question

truthfully.

Sue, unaware of Sally's conversation, asks Sam two questions.

Sue asks "Is your house number a perfect cube?". He answers.

She then asks "Is it greater than 25?". He answers again.

Sue thinks she knows where Sam lives and decides to pay him a visit.

She too is mistaken as Sam once again answered only the second

question truthfully.

If I tell you that Sam's number is less than Sue's or Sally's,

and that the sum of their numbers is a perfect square multiplied

by two, you should be able to figure out where all three of them

live.

logic/same.street.s

Sally asks Sam "Is your house number a perfect square?". He answers.

Then Sally asks "Is is greater than 50?". He answers again.

Sally thinks she now knows the address of Sam's house and decides to

visit.

Since Sally thinks that she has enough information, I deduce

that Sam answered that his house number was a perfect square

greater than 50. There are two of these {64,81} and Sally must

live in one of them in order to have decided she knew where Sam

lives.

When she gets there, she finds out she is wrong. This is not

surprising, considering Sam answered only the second question

truthfully.

So Sam's house number is greater than 50, but not a perfect

square.

Sue, unaware of Sally's conversation, asks Sam two questions.

Sue asks "Is your house number a perfect cube?". He answers.

She then asks "Is it greater than 25?". He answers again.

Observation: perfect cubes greater than 25 are {27, 64}, less

than 25 are {1,8}.

Sue thinks she knows where Sam lives and decides to pay him a visit.

She too is mistaken as Sam once again answered only the second

question truthfully.

Since Sam's house number is greater than 50, he told Sue that

it was greater than 25 as well. Since Sue thought she knew

which house was his, she must live in either of {27,64}.

If I tell you that Sam's number is less than Sue's or Sally's,

Since Sam's number is greater than 50, and Sue's is even

bigger, she must live in 64. Assuming Sue and Sally are not

roommates (although awkward social situations of this kind are

not without precedent), Sally lives in 81.

and that the sum of their numbers is a perfect square multiplied

by two, you should be able to figure out where all three of them

live.

Sue + Sally + Sam = 2 p^2 for p an integer 64 + 81 + Sam = 2 p^2

Applying the constraint 50 < Sam < 64, looks like Sam = 55 (p = 10).

In summary,

Sam = 55

Sue = 64

Sally = 81

-- Tom Smith <tom@ulysses.att.com>

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