Showing posts with label puzzle games online. Show all posts

Friday, June 12, 2015

Carpet Layer

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Walter Wall is a carpet layer. He and his two apprentices are asked by a nightclub owner to give a quote on laying carpet.

The owner indicates an oblong dance floor (figure on the right) and tells them that he wants a square of carpet adjacent each of the sides and running its entire length, making four squares in all (figure below left).

Walter asks both apprentices how many measurements must be made to calculate the total area of carpet needed in order to give a quote.
 
Sam, the slower of the two, replies that eight measurements are needed: two sides of each square.

Walter reprimands him, reminding him that these are squares and therefore have all sides the same length, and that they are in identical pairs, "So we only need to take two measurements - one side of one of the large squares and one side of one of the smaller squares".

Brian, the bright apprentice, points out that they can give the quote after taking only one measurement.


 How can the total area (that is, the sum of the areas of the four red squares) be calculated by taking just one measurement?

Carpet Layer Puzzle Solution

The only measurement that needs to be taken is the distance between opposite angles of the rectangular dance floor (figure on the left). That distance then will be squared (figure on the right) and doubled to get the sum of the areas of the 4 squares.

Tuesday, March 24, 2015

Sherwood Forest

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"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.

Monday, March 9, 2015

General Manoeuvres

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A platoon of 40 soldiers, inclusive of troopers, senior soldiers, sergeants and commander, was standing by the bank of a river. In order to cross it, they found only a small rubber rowing boat and a pair of paddles, belonging to two young boys. Due to its rather restricted size, the boat can only carry either the two boys together, or a single grown-up.

While the lieutenant - commander of the platoon - was trying to figure out the best way to organize the crossing, the radio received an urgent request: the captain wanted to know exactly how long the platoon would take to cross the river; ie how many minutes, or hours, or days were needed before the last man set his foot on the opposite bank of the river.

The lieutenant worked out that the boat, when carrying the two boys, would take 10 minutes to cross the river. One boy alone on the boat would need 5 minutes. One soldier - soldiers are not the best rowers - would take 8 minutes.

These calculations included the time taken by people to jump on board and get off the boat. After a few seconds, the officer, who had an above average IQ, sent the message with the answer to his captain.

How was the crossing organised, and how long did it take for the entire platoon to cross the river?


General Manoeuvres Puzzle Solution

The manoeuvre was conducted this way:
  1. The two boys cross to the opposite bank (10 mins)
  2. One of them stays there while the other comes back (5 mins)
  3. The boy gets off the boat, a soldier jumps on board and crosses the river (8 mins)
  4. The soldier gets off, and the boat returns with the other boy (5 mins)
This operation required 28 minutes. The sequence had to be repeated as many times as the number of men in the platoon, ie 39 more times. However, it was needed to subtract 5 minutes from from the total: when the last man of the platoon crossed, the time (5 mins) taken by the second boy to cross back must not be counted, as the last soldier had already reached the other bank of the river.

The total time was therefore [(28 * 40) - 5] = 1115 minutes, which amounts to 18 hours and 35 minutes.

Mike Horan points out that it can be done faster if you leave both boys stranded on one side with the boat on the other. The first 39 soldiers cross at 28 minutes each (1092 minutes). You then have the two boys plus the last solider on one side. The final soldier then rows across himself, hence 1092 + 8 = 1100 minutes.

Friday, January 23, 2015

Deneb

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She was an absent minded one, forever seeing things that interest her intellilect and hold her in thought for hours on end. On this particular time, she saw a curious symbol, her mental processes setting formulating a problem to solve.

deneb

Starting at any point, how can you draw the whole symbol by tracing along the lines and never tracing along the same line twice?

Deneb Puzzle Solution

deneb solution

Sunday, November 9, 2014

The Greatest Show On Earth

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"This is terrible," the customs officer shouted. "It's impossible to count all these people and animals that keep moving around constantly. I can't count the same number twice! There are zebras, lions, giraffes, horses, elephants, rhinos, tigers, cheetas, flamingos, storks, doves and hummingbirds! I can't keep them under control, I just can't!"

"Count the heads, officer," the circus owner advised. "Every animal has got a head and, as the documentation shows, I can count 112 of them."

"Count the legs, officer, that's the way," the clown said. "There are 310 legs; if you subtract the number of heads from the number of legs, you'll be able to tell how many bipeds and how many four-legged animals there are."

"Wait a second, I've just counted the legs just now, and there's 297 of them," the cook said.

"This is driving me mad," the officer muttered. "One of them tells me there are 112 heads, another one talks about 310 legs, and that mentally-disturbed cook tells me that the number of legs is only 297. How many animals, including humans, are there?

To be more precise, how many bipeds and how many four-legged animals are there? And is the cook completely mad?

The Greatest Show On Earth Puzzle Solution

There are 69 bipeds and 43 four-legged animals. If all animals were four-legged, the officer would have counted 448 legs, not 310. Obviously 138 legs are missing, hence 69 subjects are bipeds.

It's worth pointing out that, within the bipeds, there are 13 stilt birds, which include flamingos and storks. The cook must have counted them as they were standing on one leg only, so whilst he is not crazy, he almost drove the customs officer mad.