B0.40 PuzzlesSome puzzles that have amused us over the last couple of years. Some easy, some hard.
Don't be put off by the first few which are harder than most.
Sequencing I is an easy place to start.
(If you hate doing puzzles and want to laugh at people who do, then see this.)
[Slightly easier version available here.]
An example is shown below.
Prove that the big rectangle has the same property.
Extension: does this generalise to fitting together n-cuboids with the property that m<n of their sides have integer length?
Note added later: This question was answered in the affirmative by Postlethwaite in 2005, using advanced slicing and hand-waving techniques. It has now come to be known as Postlethwaite's Integer Length Cuboid Hypothesis for Any Real Dimension (PILCHARD).
How to move: iff the squares above and to the right are free, a counter can be removed and replaced by two counters, one in the square above and one in the square to the right.
Challenge: prove that it is not possible to leave the three bottom left squares empty.
Or as James would say: Which number, the answer being an integer, best fits as the next number in the sequence which begins on the next line, continues for a n lines, where n is an integer, and then terminates with the number, indeed the integer, 312211?
[Hint: a line through the equals sign, making it a not-equals sign, is not the required solution.]
[The stupidist (incorrect) answer so far: add a horizontal line through the center of the 0 and define \theta to be 3/11.]
[Hint: the answer only uses three symbols.]
[Note: however, this can be done for a square, e.g. see here.]
[The required solution does not involve a not-equals-to sign.]
It is possible to loop the rope around the stakes such that it cannot be removed by pulling it horizontally. One example of how this can be done viewed from above is:
The problem is to satisfy the above requirement in such a way that when either of the two stakes is removed, the rope can then be pulled away horizontally. (The configuration drawn above is not good enough, as the rope will still be looped around one stake even when the other has been removed.)
The second part of the problem is to achieve the same for any number of stakes - when any single stake is removed, it should be possible to remove the rope by pulling horizontally.
How to move: pick a coin and slide it, without disturbing any other coins, so that it ends up touching two other coins.
Each inmate will be wearing either a white cap or a black cap, but will not know which colour it is. You will all then stand in a long line, each man facing the back of the man in front, so that you can see the caps of the men in front of you (but not your own, nor any behind you).
The guards then ask each man what the colour of his own hat is, beginning with the prisoner at the back of the queue. A correct answer means prolonged life for that prisoner; a mistake will lead to a swift (and painful) death. The guards will continue from the back to the front of the queue, asking each inmate in turn.
You are to discuss strategy with the other prisoners beforehand. What is the best tactic to adopt in order to save as many men as possible, and consequently how many can you save?
Fifthbeard, the lowest-ranked pirate, must make a suggestion as to how to divide the money. If he can gain the support of a majority of the five (including himself) then the coins will be allocated as he suggests. If not, he will be cutlassed into tiny pieces.
If he is killed, then Fourthbeard, the pirate of the next lowest rank must make a suggestion, and if this gains the support of a majority, he will live. Otherwise he will be killed, and Thirdbeard must make a suggestion. This goes on until the (remaining) pirates reach agreement.
Each pirate values life above anything else, followed by wanting to make as much money as possible. All other things being equal for him, a pirate will choose to kill another pirate rather than allow him to live. Assuming that the pirates always think logically, what should Fifthbeard suggest in order to achieve his best possible outcome?
Each fuse burns for exactly four minutes, but they do not burn at an even rate. One fuse is attached to the explosives and lit, the explosion occurring four minutes later. The problem is that your men must leave one minute before the explosion - any later and you risk being caught in the explosion, any earlier and the enemy will have time to defuse the explosives. However, you have not watch or clock with which to measure time, just the three fuses and the explosives.
How will you light the fuses to ensure your safety?
"In the prison is a switch room, which contains two light switches labeled 1 and 2, each of which can be in either up or the down position. I am not telling you their present positions. The switches are not connected to anything.
"After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none either. Then he'll be led back to his cell.
"No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners at random. I may choose the same guy three times in a row, or I may jump around and come back.
"But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time anyone of you may declare to me, 'We have all visited the switch room.' and be 100% sure.
"If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators. You will be carefully monitored, and any attempt to break any of these rules will result in instant death to all of you"
What is the strategy they come up with so that they can be free?
First: I do not know the sum.
Second: I knew that. The sum is less than 14.
First: I knew that. However, now I know the numbers.
Second: And so do I.
What were the numbers?
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