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.

Monday, November 24, 2014

Top Secret

With his heart rate increasing steadily, James Bents (alias Lt-Colonel Ivanovic Zdanov, as far as the KGB were concerned) lined up behind the scientists who were walking towards the internal gate. Thanks to his forged documentation, he was able to pass through the two previous gates. He was aware that to get right inside the missile launch-pad, he would need to supply a password. He had been informed that the password changed daily. Only his extreme cool and many years of training enabled him to contain the fear.

The two scientists in front of him reached the gate, which was patrolled by machine-gun wielding soldiers. He strained to hear the voices of the people ahead of him in the queue.

"Twelve?" asked the guard.

"Six," replied the first scientist.

The first scientist strode through the gate as the second one walked to the guard.

"Six?" asked the guard.

"Three," replied the second scientist and walked through.

Relief and confidience spread through Bents; the method that drove questions and answers was trivial. He stepped forward.

"Nine?" asked the guard.

Brents hesitated for a split second. This was an unpredicted complication, but his arduous conditioning allowed the secret agent to remain calm and as sharp as a razorblade. "Four and a half," he answered without blinking.

Quite suddenly, the entire area was filled with floodlights. Alarm sirens broke the silence of the otherwise peaceful night. In a fraction of a second the Lt-Colonel realised his mistake. He tried to turn on his heels and run, but instantly felt the cold barrel of a machine-gun pressed against his neck.

What was the secret agent's fatal mistake?

Top Secret Puzzle Solution

The answers given were the the number of letters in the question. When asked "Nine?" the secret agent should have answered "Four".

Sunday, November 23, 2014

Luminarium Labyrinth

Luminarium is a project of British architectural bureau Architects of Air, which is traveling the world for over twenty years. During this time, it had more than five hundred exhibitions in thirty-eight countries. 

Luminarium - a giant inflatable maze with winding passages and pompous domes that resemble masterpieces of Islamic architecture, Archimedean solids and Gothic cathedrals. All this incredible diversity is transformed into an inspiring monument to light and color. The height of the dome may reach ten meters, and the entire structure covers an area of one thousand square meters.







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, November 14, 2014

A Struggle For Survival

"There are only two planets in this solar system that would offer a chance of survival to a Gxz," the geoanthropologist told the commander of the starship after having examined the data from the probe. "They are the sixth one from the sun and the eight one towards the sun."

The commander turned towards the astronavigator. "Current position?"

The navigator moved his tentacles quickly on the keyboard, "We are approaching one of the three orbits that are between the two mentioned by the anthropologist; the most internal one of the three, to be precise."

"What is this planet like," the commander asked.

"Uninhabitable," replied the geoantropologist. "The atmosphere is full of lethal gases such as oxygen. Gravity is moderate, and there's bucketloads of a mixture of hydrogen and oxygen without any silicon whatsoever; just thinking about it makes my verrucas crawl!"

"How many planets in this solar system?" asked the commander.

"Less than 12," the astrophysicist replied, peering at the instruments. "The exact number is..."

That's the end of the commander's log, found within the wreckage of the starship, and painstakingly translated. How many planets did that solar system contain?

a solar system

A Struggle For Survival Puzzle Solution

The answer is nine planets. The geoanthropologist states that one of the habitable planets is eighth toward the sun, so there must be at least 8 planets. The navigator mentions that there are 3 planets between the 6th planet and the 8th towards the sun. The only number that can satisfy both statements is the number nine. You will then realise that the navigator was talking about Earth.

Sunday, November 9, 2014

The Greatest Show On Earth

"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.

Tuesday, November 4, 2014

A Thinking Man

Professor Percent was a maths lecturer with an interest for new ways to express mathematical expressions. The traditional symbols (+, -, *, /, etc) were not enough anymore, to convey his superior numeric operations, so he had to invent new symbols, and only a superior brain would be able to understand the need for his new symbols.

The first symbol he invented was §; between two numbers, it meant that, if the first number was greater than the second, then the second should be subtracted from the first one; otherwise the two numbers should be added. Therefore 5 § 2 = 3, while 2 § 5 = 7.

The poor people that had to put up with this were, of course, his students. In the last test they were faced with:

5 ¿ 2 = 27
6 ¿ 3 = 27
8 ¿ 4 = 36

and also with:
5 ¤ 2 = 15
6 ¤ 4 = 12
3 ¤ 8 = 40

What are the meanings of the symbols ¿ and ¤?

 

Notes:
There are at least 3 different solutions for ¿.

A Thinking Man Puzzle Solution

The symbol ¿ means the difference between the number made up of the all digits of the operation, and the mirror of this last number. i.e,
5 ¿ 2 = 52 - 25 = 27
6 ¿ 3 = 63 - 36 = 27
8 ¿ 4 = 84 - 48 = 36

An alternative solution for this symbol (submitted by Alfa Chan... many thanks!) is simply the difference between the two numbers multiplied by 9. i.e,
5 ¿ 2 = (5 - 2) × 9 = 27
6 ¿ 3 = (6 - 3) × 9 = 27
8 ¿ 4 = (8 - 4) × 9 = 36

Another alternative solution submitted by Mickey Kawick... thanks!! We have x ¿ y; If x is odd, then the result is 5x + y, otherwise it's 5x - y. i.e,
5 ¿ 2 = 5 × 5 + 2 = 27 (5 is odd, so we add the 2)
6 ¿ 3 = 6 × 5 - 3 = 27 (6 is even, so we subtract the 3)
8 ¿ 4 = 8 × 5 - 4 = 36 (8 is even, so we subtract the 4)

The symbol ¤ means the difference between the two numbers multiplied by the larger of the two numbers. i.e,
5 ¤ 2 = (5 - 2) * 5 = 3 * 5 = 15
6 ¤ 4 = (6 - 4) * 6 = 2 * 6 = 12
3 ¤ 8 = (8 - 3) * 8 = 5 * 8 = 40

An alternative solution for this symbol, as submitted by Jeff Schall (many thanks!), is the difference between the square of the bigger of the two numbers and their product. i.e,
5 ¤ 2 = (5 ^ 2) - (5 * 2) = 25 - 10 = 15
6 ¤ 4 = (6 ^ 2) - (6 * 4) = 36 - 24 = 12
3 ¤ 8 = (8 ^ 2) - (3 * 8) = 64 - 24 = 40