Showing posts with label brain teasers. Show all posts

Monday, August 31, 2015

A Farmer's Good Fortune

A farmer from a small community is out of money. After a mysterious desease spread among and killed his lifestock, he now needs to quickly make up for the lost animals. He needs a whole grand.
Knowing that the bank won't lend him any money, he pays a visit to the local loan shark. The outlaw, who's known to have a bit of an obsession with puzzles, proposes a deal.

With the $1,000 he gets, the farmer has to be able to buy a combination of cows, pigs, and sheep, to total exactly 100 heads of lifestock. The combination has to include at least one cow ($100 each), one pig ($30 each), and one sheep ($5 each). The total amount of money spent for the 100 animals has to equal exactly $1,000.

farmer's fortune riddle

If the farmer manages to accomplish the task, he'll have to return the money with a "friendly" interest rate. Otherwise, he'll get the normal rate, and the threat of a broken pinkie...

How many of each kind of livestock did the farmer buy?

A Farmer's Good Fortune Puzzle Solution

Here's one combination:

 94 sheep =  $470
  1 pig   =   $30
  5 cows  =  $500
-----------------
100 heads = $1000

Are there anymore?

Friday, August 21, 2015

Switching Logic

bulbs logic game
You are in a basement. In front of you are three light switches, all in their OFF state. These switches are connected to three lightbulbs that are on a wall in the attic, so there's no chance you can see them unless you climb the stairs all the way up. There is a 1-to-1 mapping between the switches and the bulbs.

You are given plenty of time to play around with the switches in the basement, where you can put each individual switch in either its ON or OFF state.

However, you can only go upstairs once to check on the state of the bulbs!

Once you've gone upstairs and checked on the bulbs, you must be able to tell with 100% certainty which bulb is connected to which switch, without having to go down again.

How can you tell which switch is connected to each bulb?

 

Notes:
  • You can't put a switch halfway between ON and OFF, hoping that this would make the bulb flicker like a bad neon light...
  • You can't control the switches from a distance, eg with a string or whatever other form of remote control.
  • You can't have anyone cooperating with you on the other floor, and that includes your dog who knows how to bark once for a 'yes' and twice for a 'no'.

Switching Logic Puzzle Solution

Turn switch #1 ON. After about five minutes or so, turn switch #1 OFF and turn switch #2 ON.
Then go upstairs and check on the bulbs.

The one ON is obviously #2. The other ones are OFF, but one of them should be very hot by having been ON for five minutes. That's #1, and the remaining bulb is #3.

Thursday, August 6, 2015

U2 Gig

u2 gig logical game
Bono, The Edge, Adam, and Larry must cross a narrow darkened bridge in order to reach the stage where they are due to play in 17 minutes.

Unfortunately they only have one torch between them which must be used to cross the bridge safely and may not be thrown, only carried across the bridge.

The bridge will hold up to two band members at any time.

Each member crosses at their own pace and two members must go at the slower members pace.

Bono can cross in one minute, The Edge in two, Adam in five, and Larry in ten.

How do they make it in time?

U2 Gig Puzzle Solution

The trick is to get the two slowest people to cross at the same time. One solution is...
  • Bono and Edge cross the bridge for which they take 2 mins (Total time = 2)
  • Then Bono comes back with the torch (Total Time = 2 + 1 = 3)
  • Then Adam and Larry cross the bridge (Total time = 3 + 10 = 13)
  • Then Edge comes back (Total = 13+2 = 15)
  • Then both Bono and Edge cross the bridge (Time = 15+2=17)

Wednesday, July 22, 2015

The Diagram

Bob looked up from his book and noticed that Joe was staring at the VCR clock, holding a pencil and a pad of paper. He knew the clock was set correctly, having set it himself. But he noticed that Joe would occasionally write something down.

Finally Bob's curiosity got the best of him: "What are you doing, Joe?"

"Just another minute" came the reply.

"How long are you going to stare at the clock like that?" As Bob finished he noticed the clock advance a minute.

At this, Joe started scribbling some more until he produced this diagram:
8
6 8
7
4 9
7
"Why, just ten minutes, of course", Joe beamed as he showed the diagram to Bob.

What was Joe doing, and what does his diagram mean?

The Diagram Puzzle Solution

Joe counted how many times each of the 7 LEDs lit up for all 10 digits...
 _
|_|
|_|

Tuesday, June 2, 2015

Loadsa Coins!

conis logical game
Laid out neatly in front of you as far as you can see are coins, coins and more coins. "That's a lot of coins," you think to yourself.

Greed sidles up to you and whispers, "Actually, there are an infinite number. And it can all be yours!"

Upon seeing your raised eyebrow, he continues. "The rules are simple. 20 of the coins are heads, the rest are tails. All you have to do is split all the coins into just two groups so that the number of heads is the same in both groups."

Seeing the obvious difficulties, you begin to protest. "But..."

"You can turn over as many coins as you like," he interrupts. "And the groups do not have to have the same number of coins in them."

"Oh ok. Can I start now?"

"Sure, you have one day to complete this task. You can start as soon as I blind you for one day."
"How can I get through it all in a day...! What do you mean blind me...!?"

Although you cannot see him, you are quite certain that he is grinning.

You cannot see. You cannot feel the difference between head nor tail. How can you accomplish this task with certainty in a single day?

Loadsa Coins! Puzzle Solution

You reach out and pick any 20 coins, and turn each of them over. The 20 coins is one group, and the remaining coins the other.

"Bugger," says Greed. "Why didn't I think of that?"

"It's so simple, yet so difficult to think of," you say. "Now can I have my eyesight back? Hello? Hello......!?"

Tuesday, April 28, 2015

Socks And Shops

socks colors puzzle
Sarks and Mencer's was a brand new shop built in the basement floor of a shopping centre. To mark their opening day, they had many special offers. The most popular was the buy ten pairs of socks for half price. Each pair of socks were attached together with a plastic clip that doubled as a miniture hanger whilst on display. There was only a single size and only ten colours to choose from, which is why both Andrew and Bob each chose one of each colour. Each armed with ten pairs of differently coloured socks, they were moving towards the checkout when there was a power failure. They bumped into each other and dropped all twenty pairs of socks.

Without being able to see the colour of the socks, how could they divide them so that they both got 10 pairs of uniquely coloured socks each?

Socks And Shops Puzzle Solution

For each of the twenty pairs of socks, they unclipped it and gave one sock each.

Monday, April 13, 2015

TV Show

3 doors puzzle
The host of the show pointed at three doors. He claimed that, behind one of the doors, a brand new sportscar was awaiting a lucky winner. The other two doors, he warned, did not lead to any prize.

The contestant picked the first door as his guess. At that point, the host walked to the third door and opened it. The door led to no prize, which is something the host knew perfectly well. He then gave the chance to the contestant to switch and pick the second door, if he so wished, or to stick to his first choice and stay with the first door.

Did the contestant have a greater chance of winning the car, by sticking with the 1st door, or by switching to the 2nd door? Or were the chances equal?

 

Notes:
The contestant knew three things: first, that the host doesn't want him to win; second, that the host was going to open one of the doors; and third, that the host would never open the door picked by the contestant himself... regardless of where the prize really is.

TV Show Puzzle Solution

This puzzle isn't particularly new, but it became very well known in 1990, thanks to the person that allegedly had "the highest IQ ever recorded" -- 228, according to The Guinness Book of WorldRecords. Miss Marilyn vos Savant (yes, it was a female) wrote a solution to this puzzle on her weekly column, published in a popular magazine. This solution led to waves of mathematicians, statisticians, and professors to very heated discussions about the validity of it.

The contestant, according to vos Savant, had a greater chance of winning the car by switching his pick to the 2nd door.

She claimed that by sticking to the first choice, the chances of winning were 1 out of 3, while the chances doubled to 2 out of 3 by switching choices. To convince her readers, she invited her readers to imagine 1 million doors instead of just 3. "You pick door number 1," she wrote, "and the host, who knows what's behind every door, and doesn't want you to win, opens all of the other doors, bar number 777,777. You wouldn't think twice about switching doors, right?"

Most of her readership didn't find it as obvious as she thought it was... She started receiving a lot of mail, much of it from mathematicians, who didn't agree at all. They argued that the chances were absolutely equal, whether or not the contestant switched choice.

The week after, she attempted to convince her readers of her reasoning, by creating a table where all 6 possible outcomes were considered:

Door 1 Door 2 Door 3 Outcome
(sticking to door 1)
Car Nothing Nothing Victory
Nothing Car Nothing Loss
Nothing Nothing Car Loss
Door 1 Door 2 Door 3 Outcome
(switching to other door)
Car Nothing Nothing Loss
Nothing Car Nothing Victory
Nothing Nothing Car Victory

The table, she explained, shows that "by switching choices you win 2 out of 3 times; on the other hand, by sticking to the first choice, you win only once out of of 3 times".

However, this wasn't enough to silence her critics. Actually, it was getting worse.

"When reality seems to cause such a conflict with good sense," wrote vos Savant, "people are left shaken." This time, she tried a different route. Let's imagine that, after the host shows that there's nothing behind one of the doors, the set becomes the landing pad for a UFO. Out of it comes a little green lady. Without her knowing which door was picked first by the contestant, she is asked to pick one of the remaining closed doors. The chances for her to find the car are 50%. "That's because she doesn't have the advantage enjoyed by the contestant, ie the host's help. If the prize is behind door 2, he will open door 3; if it is behind door 3, he will open door 2. Therefore, if you switch choice, you will win if the prize is behind door 2 or 3. YOU WIN IN EITHER CASES! If you DON'T switch, you'll win only if the car is behind door 1".

Apparently, she was absolutely right, because the mathematicians reluctantly admitted their mistake.

Thursday, March 19, 2015

Golf

Jack, Levi, Seth, and Robert were, not necessarily in this order, a Stock Broker, a Musician, a Doctor, and a Lawyer. They drove, also not necessarily in order, a Porsche, a Ferrari, a Cadillac, and a Corvette.

The Stock Broker, was remarking to no one in particular one day, while finishing up a round of golf which involved all four friends, that he found it curious that Jack and the Lawyer each wanted to buy the Corvette, but that the Musician didn't because he preferred his Porsche. After the game was over, Seth offered to buy a round of sodas for the Doctor, for the owner of the Cadillac, and for the owner of the Corvette. Levi, who was beaten by the Stock Broker, was in a bad mood, and so he declined Seth's offer and left without joining the others in the club house.

What car did each person own, and what were their respective occupations

Golf Puzzle Solution

The clues given in the puzzle are:
  • The Stock Broker is not Levi.
  • Jack is not the Lawyer; neither Jack nor the Lawyer owns the Corvette.
  • The Musician owns the Porsche; therefore no one else owns the Porsche and the Musician owns no other car.
  • Seth is not the owner of the Cadillac, the Corvette, nor is he the Doctor; also, the Doctor does not own either the Cadillac or the Corvette either.
By Reasoning:
  • The Doctor, by elimination, must own the Ferrari; the Lawyer must own the Cadillac; the Stock Broker must own the Corvette.
  • Tne Stock Broker owns the Corvette. Neither Seth nor Jack own the Corvette, so the Stock Broker must be Robert; since the Stock Broker drives the Corvette, then Robert drives the Corvette.
  • The Lawyer owns the Cadillac. Seth does not own the Cadillac; therefore Seth is not the Lawyer; therefore Seth is the Musician; the Musician owns the Porsche; therefore Seth owns the Porsche.
  • By elimination, Jack is the Doctor; the Doctor owns the Ferrari; therefore Jack owns the Ferrari.
  • By elimination, Levi owns the Cadillac.
The Solution:
  • Seth is the Musician and owns the Porsche.
  • Levi is the Lawyer and owns the Cadillac.
  • Jack is the Doctor and owns the Ferrari.
  • Robert is the Stock Broker and owns the Corvette.

Monday, March 9, 2015

General Manoeuvres

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, February 27, 2015

Snake

How are we to seat them within our only 8 remaining spaces?" frowned the manager of the theatre. Everyone was well aware of the problems of these eight people. They came as a group, but each person loathed one or two of the other people and could not stand to be near them.
The assistant manager gestured to the plan of seats remaining. "I have made eight counters to represent each of the people in this group. I have numbered them logically. Person 1 and 8 only hate 2 and 7 respectively. All the others hate the numbers on either side of them." He rubbed his chin in thought, "the only problem is how can we arrange them so that no-one is right next to anyone they hate?"

Place the counters 1 to 8 in the grey squares above such that no two consecutive counters are adjacent to one another horizontally, vertically nor diagonally.

Snake Puzzle Solution

Counters 1 and 8 need to be in the middle since they each have only one number that they cannot be neighbours with. Hence 7 and 2 must go in the sidewings, and the rest is trivial.

3 5
7 1 8 2
4 6

Sunday, January 18, 2015

Visit At The White House

"This," started the guide, "is the Buttons Room."

The Druggar of Bongo Ghango - the chief of a large country by the River Ghango - looked around. "It's a nice room, but where are the buttons? The only ones I see are, ahem, the ones on your shirt."
"Your Highness. The buttons, which could start a nuclear armageddon in a matter of seconds, are there, behind that panel," replied the guide, while pointing at a large panel at the end of the room.
"How can that be??? Why? Anybody - a madman, for example - could come here and press those terrible buttons?"

"Your Highness, it is very safe actually: for every button there is a slot, where a magnetic card must be inserted to activate the corresponding button. No card, no button. To launch the missiles, all buttons must be activated and pressed, and only a handful of people have the magnetic cards, and each card contains a different code from all the other ones."

"But it's the same thing! Any of these persons could go completely ballistic and start a nuclear war."
"In that case, the only dangerous man is the President of the United States, as he is the only person that holds all the codes, which would allow him to press all the buttons. The other people that hold some codes are the Vice-President of the United States, the President of the Senate, the Secretary of State, the Chief of the Armed Forces, and the Dean of Harvard University. Each of these gentlemen holds an incomplete set of magnetic cards, and the distribution of the codes is such that, if the President of the United States is not available, the entire set of buttons can be activated by the Vice-President, together with anyone of the other four men. If both the President and the Vice-President of the United States are unavailable, the buttons can still be activated by any three of the other four men. Therefore, to launch the missiles, it is needed either the President, or the Vice-President plus anyone of the other four men, or any three of the other four men."

"What if someone tried to press randomly many buttons, one after the other?" asked the Chief.
"Nothing would happen with the missiles, but the room would fill up with a narcotic gas, and an alarm would alert the guards and the CIA."

"So, how many buttons are there, and how are they distributed between the Vice-President and the other four personalities?"

And that's the question we'll ask the reader: what is the minimum number of buttons, and how are they distributed?

Visit At The White House Puzzle Solution

There are 7 buttons. The magnetic cards, as held by the six persons, and marked with an X, are distributed as follows:
Person Buttons
President X X X X X X X
Vice President X X X X X X -
President of Senate X X X - - - X
Secretary of State X - - X X - X
Chief of Armed Forces - X - X - X X
Dean of Harvard - - X - X X X

Tuesday, December 9, 2014

Drama Galore!

It was a beautiful day, perfect for a stroll. After leaving the cars at the edge of the woods, four couples moved towards the river, reaching its bank after a two-mile walk. The restaurant where they intended to dine was on the other side, partially hidden by trees.

But even on a perfectly planned day, evil could come and spoil it. During the stroll, Albert quietly told Amanda that she shouldn't have dressed quite so promiscuously, whilst she replied that he could have done a better job in refraining from making his oh-so-kind compliments to the other three girls. Bernard whispered menacing words at Barbara The Easy Flirt (as he called her then), and Barbara told him that his relationship with the other girls was of dubious morality. Simillar sorts of exchanges happened between Charles and Corinna, and between Douglas and, err, Diana. Reaching the river and seeing the flowing water did little to tame the souls. On the contrary, when the eight friends noticed that instead of the large boat that would carry them all over to the opposite bank, there was only a little boat that would carry no more than two persons at once, the irritation grew to the point that everybody started arguing with everybody else.

The river was about one hundred yards wide, with a small island in the middle. None of the four men were keen on leaving his girlfriend alone with one or more of his other male friends. On the other hand, the women found out that they could only agree on one point: none of their boyfriends should be alone on the boat when one of the girls, excluding his girlfriend, was all alone on any of the two banks or on the island.

Once the tempers calmed, Bernard and Douglas forumlated a plan involving many trips.

How many trips would it take to ferry everyone across whilst still adhering to the wishes of all the people?

 

Notes:
  • In case you haven't guessed, the couples are, Albert and Amanda, Bernard and Barbara, Charles and Corinna, and Douglas and Diana.
  • No woman should stay on one of the banks, on the island, or on the boat, in the company of one or more other men and without her boyfriend
  • No man should be alone on the boat when one of the girls, except his girlfriend, was all alone on one of the two banks or on the island

Drama Galore Puzzle Solution

They needed seventeen trips, frenquently using the island in the middle of the river as temporary destination. If we denote the names of the men with their initial in upper case, and the names of the women with their initial in lower case (by complete chance, the initials of the men are the same as the initials of their girlfriends), we would get the following table:
Trip # Departure bank Direction Island Direction Arrival bank
1 ABCDcd -
=>
ab
2 ABCDbcd
<=
- a
3 ABCDd
=>
bc a
4 ABCDcd
<=
b a
5 CDcd b
=>
ABa
6 BCDcd
<=
b Aa
7 BCD
=>
bcd Aa
8 BCDd
<=
bc Aa
9 Dd bc
=>
ABCa
10 Dd abc
<=
ABC
11 Dd b
=>
ABCac
12 BDd
<=
b ACac
13 d b
=>
ABCDac
14 d bc
<=
ABCDa
15 d -
=>
ABCDabc
16 cd
<=
- ABCDab
17 - -
=>
ABCDabcd

Saturday, November 29, 2014

Galactic Expedition

When the scientific community predicted that the Sun would explode into a supernova in ten years time, destroying the entire Solar System, the Earthlings duly prepared for the Doomsday event. The first intergalactic spaceship was already being built, even before that prediction of doom. A sample of the human population was going to travel to the Andromeda Galaxy, where a planet in the BSC14823 solar system was believed to be able to sustain human life. It was hoped that on this planet, the human race would survive and re-establish itself as the superior being in the universe.

The person that would navigate the ship would have to rely on his/her own mathematical abilities, since obviously there could be no assistance from Earth after the explosion. The journey was predicted to last for at least 30 years, so the selection of this person was extremely tough. Only the very finest mathematical and logical mind would be able to successfully navigate the spaceship through intergalactic travel. A test to find this person was issued, and the best mathematicians of the planet competed.

spaceship puzzle

Competition was feirce, for this person would become the first hero of the new world. Of all canditates, only a young woman and a young man reached the very final stage of the selection process. The chairman of the Special Selection Board explained the final question, "There are three different integers, A, B, C, where A × B × C = 900, and A > B > C. One of you will receive either A + B or A + C, but you will not know which one of the two sums it will be. The other candidate will be given the number B. You will take turns, and the winning candidate will be the one that can tell us what the 3 numbers are."

The young man, named Wadzru, was asked to start first. "I don't know," was his answer. Then it was the young woman's turn. Her name was Zaxre, and her answer was also "I don't know." Wadzru's second turn was another "Don't know," which is the same answer given by Zaxre on her second turn. They went on answering "I don't know" for a certain number of times, until Wadzru suddenly grabbed the pen and started writing down the answer. As soon as he started writing, Zaxre cried a "YES!" and also started writing down the answer.

The selection board was confused. Wadzru did indeed answer first, but only because he was one turn ahead. Zaxre also answered correctly when it was her turn. At that moment a young mathematician entered the room. The chairman of the board called him over, and explained the mathematical problem to him. After a short pause, the man said, "If I understand the problem correctly, sir, then the two candidates must have given the correct answer on their fourth turn, that is when each of them were answering for their fourth time. At this point, I can also tell you what the three numbers are. However, there are two different answers, depending on who was given the first chance to answer: whether it was the candidate who was given number B, or the candidate who was given the sum A + B or A + C."

The board were so impressed by this latecomer that he was nominated as the commander of the expedition.

What are the two different combinations of the three numbers?

Galactic Expedition Puzzle Solution

The table below shows all possible combinations (32 of them) of three different integers, the product of which is 900. Next to each combination there is the result of the sum A+B, then the result of the sum A+C, and finally the the number B for that particular combination:
Combination # Factors A+B A+C B
1 450 * 2 * 1 452 451 2
2 300 * 3 * 1 303 301 3
3 225 * 4 * 1 229 226 4
4 180 * 5 * 1 185 181 5
5 150 * 6 * 1 156 151 6
6 100 * 9 * 1 109 101 9
7 90 * 10 * 1 100 91 10
8 75 * 12 * 1 87 76 12
9 60 * 15 * 1 75 61 15
10 50 * 18 * 1 68 51 18
11 45 * 20 * 1 65 46 20
12 36 * 25 * 1 61 37 25
13 150 * 3 * 2 153 152 3
14 90 * 5 * 2 95 92 5
15 75 * 6 * 2 81 77 6
16 50 * 9 * 2 59 52 9
17 45 * 10 * 2 55 47 10
18 30 * 15 * 2 45 32 15
19 25 * 18 * 2 43 27 18
20 75 * 4 * 3 79 78 4
21 60 * 5 * 3 65 63 5
22 50 * 6 * 3 56 53 6
23 30 * 10 * 3 40 33 10
24 25 * 12 * 3 37 28 12
25 20 * 15 * 3 35 23 15
26 45 * 5 * 4 50 49 5
27 25 * 9 * 4 34 29 9
28 30 * 6 * 5 36 35 6
29 20 * 9 * 5 29 25 9
30 18 * 10 * 5 28 23 10
31 15 * 12 * 5 27 20 12
32 15 * 10 * 6 25 21 10

If the first question was asked to the person (Wadzru) that was given A+B or A+C and he answers "don't know", that means that the number he received appears in the A+B and A+C columns more than once. On the other hand, if that number appears in those columns only once, then the solution would be found immediately, as that number would relate to only one of the 32 combinations.

Therefore the first candidate was given one of the following numbers: 65, 61, 37, 65, 29, 28, 27, 25, 23. These numbers appear more than once in columns A+B and A+C, and because of this, they do not yet allow to find the correct combination. But we can now discard, from the 32 combinations, the ones where the 9 numbers listed above are not included in columns A+B and A+C. The combinations left were then:
Combination # Factors A+B A+C B
9 60 * 15 * 1 75 61 15
11 45 * 20 * 1 65 46 20
12 36 * 25 * 1 61 37 25
19 25 * 18 * 2 43 27 18
21 60 * 5 * 3 65 63 5
24 25 * 12 * 3 37 28 12
25 20 * 15 * 3 35 23 15
27 25 * 9 * 4 34 29 9
28 30 * 6 * 5 36 35 6
29 20 * 9 * 5 29 25 9
30 18 * 10 * 5 28 23 10
31 15 * 12 * 5 27 20 12
32 15 * 10 * 6 25 21 10

The second candidate (Zaxre) followed the same reasoning, so she then knew that these were the only possible combinations left, after Wadzru gave his answer. The situation for Zaxre was the same though: if the number she was given appeared only once in column B, then the right combination would be the one containing that number; if the number appeared more than once in column B, then the solution would not yet be within reach. And this is what happened, since she answered "don't know". But the elimination of certain combinations could go on nevertheless, because after she said "don't know", combinations # 11, 12, 19, 21, 28 were automatically discarded. Then it was Wadzru's second turn, and the current situation was:
Combination # Factors A+B A+C B
9 60 * 15 * 1 75 61 15
24 25 * 12 * 3 37 28 12
25 20 * 15 * 3 35 23 15
27 25 * 9 * 4 34 29 9
29 20 * 9 * 5 29 25 9
30 18 * 10 * 5 28 23 10
31 15 * 12 * 5 27 20 12
32 15 * 10 * 6 25 21 10

At this point, the first contestant answered again "don't know", and automatically discarded combinations 9 and 31. Then the second candidate, after following, again, the logical reasoning of Wadzru, answered "don't know", therefore discarding combinations 24 and 25.The third turn started with the first contestant facing the following combinations:
Combination # Factors A+B A+C B
27 25 * 9 * 4 34 29 9
29 20 * 9 * 5 29 25 9
30 18 * 10 * 5 28 23 10
32 15 * 10 * 6 25 21 10

Wadzru, by answering "don't know" on his third turn, discarded the only combination with no alternatives, ie 30, so Zaxre was presented only with combinations 27, 29, 32; her answer "don't know" left only two combinations: 27 and 29. Then it was the first contestant again, and he was faced with the following situation:
Combination # Factors A+B A+C B
27 25 * 9 * 4 34 29 9
29 20 * 9 * 5 29 25 9

If the first contestant now answered "don't know", then the number he was given must have had to be 29; but since he was given number 25, he was able to know the correct combination requested: 20*9*5. When Zaxre saw that Wadzru had the answer, she was also able to find the solution, because she had also been able to follow the selection process. For her, it was only possible to give the answer on her 4th turn, because if Wadzru could have been able to come up with the answer on his 3rd turn (when the numbers appering just once in columns A+B and A+C were more than one), then Zaxre would not have been able to know which of the combinations contained the number given to her opponent.

If instead, the question was first asked to Zaxre, who was given number B, then after her first "don't know" it would have been possible to discard from the table (the one with all 32 combinations) combinations 1, 11, 12. After her opponent also said "don't know", the combinations to be discarded would have been 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23 and 26, leaving the table looking like this:
Combination # Factors A+B A+C B
19 25 * 18 * 2 43 27 18
24 25 * 12 * 3 37 28 12
25 20 * 15 * 3 35 23 15
27 25 * 9 * 4 34 29 9
28 30 * 6 * 5 36 35 6
29 20 * 9 * 5 29 25 9
30 18 * 10 * 5 28 23 10
31 15 * 12 * 5 27 20 12
32 15 * 10 * 6 25 21 10

For the second turn, the "don't know" answer of the contestant answering first (Zaxre in this scenario) would have automatically discarded combinations 19, 25 and 28, and the "don't know" from her opponent would have further discarded another one: 31. For the third turn, the remaining possible combinations are: 24, 27, 29, 30 and 32. If Zaxre could not answer yet, then combination 24 would be discarded, while Wadzru's "don't know" would discard combination 30. The situation would now be:
Combination # Factors A+B A+C B
27 25 * 9 * 4 34 29 9
29 20 * 9 * 5 29 25 9
32 15 * 10 * 6 25 21 10

If Zaxre would now be able to know the solution, she would have been given, as B, number 10 (combination # 32), resulting with the product 15*10*6. But her opponent would also be able to answer then, thanks to Zaxre being able to find the solution. This scenario, like the previous one with Wadzru starting, where both opponents are able to find the solution at the same time, can only happen on the fourth turn. It's worth knowing that a fourth "don't know" from the candidates would have led to the problem being left unsolved.

Wednesday, November 19, 2014

Killing Some Time…

  1. The only animals in this house are cats.
  2. Any animal that loves watching the moon is tamable.
  3. When I hate an animal, I stay away from it.
  4. No animal is carnivorous, unless it moves at night.
  5. No cat avoids killing mice.
  6. I do not own any animal, except the ones that live in this house.
  7. Kangaroos are not tamable.
  8. No animal, except the carnivorous ones, kills mice.
  9. I hate animals that I do not own.
  10. All animals that move at night love watching the moon.
From this sequence of sentences, which are based on the foundations of logic, it is possible to reach a conclusion deduced from all 10 sentences.

What is the conclusion?

Killing Time Puzzle Solution

The logical conclusion must necessarily be: I avoid kangaroos. Based on the 10 statements, it is possible to deduce the following conclusions:
Deduction # From statements # Deduction
11 1 and 5
All animals in this house kill mice
12 8 and 11
All animals in this house are carnivorous
13 4 and 12
All animals in this house move at night
14 6 and 13
All animals I own move at night
15 10 and 14
All animals I own love watching the moon
16 2 and 15
All animals I own are tamable
17 7 and 16
I do not own any kangaroo
18 9 and 17
I hate kangaroos
final 3 and 18
I avoid kangaroos

Friday, October 10, 2014

Faulty Batch

A little nation in Antarctica has its gold coins manufactured by eight different European companies. The Treasury Minister and his secretary were examining samples just delivered from the eight companies.

"How much should these coins weigh?" the Minister asked.

"Ten grams each, Sir."

"At least one of these coins - this one - is lighter than the others," said the Minister. "Let's check."

He put the coin on the scale, which showed that the coin weighed only nine grams. A bunch of coins, untidily placed on a tray, were frantically searched by the Minister and his secretary. Within the bunch, they found a handful of coins that also weighed one gram less than they should. The two men looked at each other; obviously, one of the manufacturing companies was producing coins with the wrong weight.

"Most of the coins are still packed in the plastic wrappers. It should be easy to tell which company is producing the faulty batch," said the secretary.

The two men placed eight packs of coins on the table, one pack from each company.

"How tedious," sighed the Minister. "Do we really have to use this scale eight more times, just to find the faulty batch of coins?"

"That won't be necessary, Sir," grinned the secretary. "We can find the lighter coins by using the scale only once."

How would they do it?



Notes:
By using the scale once, it means that only one reading can be taken after all the coins to be weighed are placed onto the scale. ie, you cannot read the values as you place the coins on -- that would make the puzzle too easy!

Faulty Batch Puzzle Solution

The secretary placed on the scale 1 coin from the first batch, 2 from the second, and so on until he put 8 from the eighth batch.

If all coins weighed 10 grams each, then the weight displayed on the scale should have been 360 grams ((1 + 2 + ... + 8) × 10). But, since one batch of coins weighs less, the difference between 360 grams and the weight displayed on the scale should point us to the faulty batch. For example, if the faulty batch was the fifth one, then the total weight displayed on the scale would be 355 grams. Or if it was the seventh batch, the weight would have been 353 grams, ie 7 grams less than the theoretical total weight of 360 grams.

An 'optimisation' on this solution is to omit the 8 coins from the eighth batch. In this case, the maximum weight of the coins would be 280 grams, and if it equals 280, then the eighth batch is the faulty one. Thanks to Denis Borris for this observation.

By using the same logic, one could omit the coins from any one of the other batches, instead of the eighth one. For example, if we omit the fourth batch, we'll be left with a theoretical 320 grams and, if it is indeed the total weight, then we will know that the fourth batch was the faulty one. Thanks to Glen Parnell for noticing this.

Monday, September 15, 2014

Mizar

mizar
The bells of the clock tower signal six o'clock (with 6 loud rings) in 5 seconds.

How long will it take for the bells to signal twelve o'clock (midday)?



Notes: It is not 10 seconds!

Mizar Puzzle Solution:

11 seconds. The 5 seconds needed to signal six o'clock are the 5 silent intermissions between rings. At twelve, the 12 rings are interleaved by 11 silent intermissions, which need 11 seconds to be executed.

Friday, June 13, 2014

The Missing Square Puzzle Paradox

The missing square puzzle is an optical illusion used in mathematics classes to help students understand geometrical figures. It depicts two arrangements made of similar shapes in somewhat different configurations.

The key to the puzzle is the fact that neither of the "triangles" is truly a triangle, because what appears to be the hypotenuse is bent. In other words, the "hypotenuse" does not maintain a consistent slope, even though it may appear that way to the human eye. So, with the bent hypotenuse the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square. Note the grid point where the red and blue triangles in the lower image meet and compare it to the same point on the other figure; the edge is slightly under the mark in the upper image, but goes through it in the lower. A very fascinating and interesting geometrical puzzle.