Friday, October 31, 2014

Scariest Halloween Pumpkins

Just take a look at these terrible and funny at the same time the creations of artist John Neil. The technique of carving the pumpkin is amazing, as well as the artistic vision that reflects the whole spirit of this beautiful holiday - Halloween.










Thursday, October 30, 2014

Three Divine Comedians



divine comedians puzzle

As Dante was reaching River Styx on his way back from the Underworld, he was expecting to hitch a ride back on Charon's boat, and then go to pay a visit in Purgatory. As things went, Charon turned out to be a rather nasty fellow. Instead of giving Dante a nice, hassle-free trip back across the damned river, he called his three best buddies. Dante looked at them. The monsters were pretty ugly overall.

"You see, Dante, nothing is free in life - or death - except suffering," said Charon. "My three friends here, although they all look repulsive, are rather peculiar: one of them always tells the truth, another one always lies, and the last one is a bit of a lunatic: sometimes it tells the truth and sometimes it lies. You have a total of three questions you can ask them to find out which one is which. I'll take you on the other side of the Styx if and only if you can tell me, without a shadow of a doubt, which one tells the truth, which one is the liar, and which one is the lunatic one. Oh, I almost forgot to tell you - silly me - they can only answer yes or no... isn't life great?"

What three questions will enable Dante to cross the River Styx?

 

Notes:
  • Each question is directed to, and answered by, only one creature.
  • The creatures themselves know who is the truth-teller, who is the liar, and who is the lunatic.
  • The solution does not rely on asking them questions that they are not able to answer due to uncertainty. eg, asking the liar or the truth-teller to predict whether the lunatic will say yes or no to a given question.

Three Divine Comedians Puzzle Solution

Dante figured that first of all, he had to find out which of the three monsters was the lunatic one. Let's call the three monsters A, B, C.

In order to do so, he asked one of them (let's say monster A for this example) a question like: "If I asked a question to monster B, would I stand a greater chance of obtaining a truth than if I asked the same question to monster C?"

The possible combinations (where + means the monster that tells the truth, - means the monster that lies, x means the lunatic monster) are:
Answer given Monster A Monster B Monster C
No + - x
Yes + x -
No - + x
Yes - x +
Yes/No x + -
Yes/No x - +

If the answer was a "Yes", then it was safe to say that x couldn't have been monster C; on the other hand, if the answer was a "No", then x couldn't possibly have been Monster B.

Once that Dante obtained this information, he used his second question to find out whether the monster that is definately NOT the lunatic one was + or -, and the easiest way of achieved this was by asking a question with an obvious answer such as, "Is 2 an even number?" After finding out whether the monster is the liar or the sincere one, the third question was used to resolve the other two monsters; a question like "Is monster A the lunatic one" did the trick.

Saturday, October 25, 2014

Barrels 'O' Fun

In the basement of the Italian "cantina", there are 3 small, irregularly-shaped, wine barrels: a 12-litre one, full, and two empty ones, which can contain up to 7 and 5 litres.

Without using any additional tool, how can you get exactly 6 litres of wine in the 7-litre barrel, and have 6 litres left in the 12-litre barrel?

Barrels 'O' Fun Puzzle Solution

There are multiple ways of solving this. One way is given below, and it's probably the fastest one. Each set of 3 numbers separated by hyphens is the amount of wine (in litres) in the 3 barrels after each "pouring operation". The 3 barrels are always in the same order: 12, 7, and 5 litres.
  • 12-0-0
  • 5-7-0
  • 5-2-5
  • 10-2-0
  • 10-0-2
  • 3-7-2
  • 3-4-5
  • 8-4-0
  • 8-0-4
  • 1-7-4
  • 1-6-5
  • 6-6-0.

Monday, October 20, 2014

Three Camels

Three young men travelled across the desert toward the tent of The Great Sage, seeking precious advice.

The eldest of the three moved in front of The Great Sage, who was meditating, and said, "God bless You, Great Sage! Our Father, before dying, left us these camels, and it is his will that I should have a half of the herd, my brother Ali one third, and my brother Ismail one ninth. We've tried, Glorious Sage, we have divided the camels and divided them again until the void opened before us. Help us, Magnificent Sage, we are not gifted with your superior intellect!"
camels riddle


The Great Sage asked the pleading man "How many camels are there?"

"Seventeen, may God bless You!", was the answer.

The Great Sage smiled.

How were the camels divided, strictly observing the fatherly will and without butchering any of them?

Three Camels Puzzle Solution

The Great Sage added his own camel to the other seventeen. He then gave 9 camels (one half of 18) to the eldest of the three, 6 camels (one third) to Ali, and 2 camels (one 9th) to Ismail. Then took his own camel back and sat in front of the tent, thanking God for His generosity.

Another way to explain the solution, perhaps in a more mathematical manner, is the following, courtesy of Gopalakrishnan Thirumurthy.
The three sons are assigned their shares: 1/2, 1/3, and 1/9. The sum of their shares is 1/2 + 1/3 + 1/9 = 17/18. Out of 18 camels, 17 of them are left by their father. So,
  • #1 gets 1/2 of 18 = 9
  • #2 gets 1/3 of 18 = 6
  • #3 gets 1/9 of 18 = 2
9 + 6 + 2 = 17.

Wednesday, October 15, 2014

Cheers To Statistics

Two and a half artists spend two and a half hours painting two and a half models on two and a half canvases.
How many artists are necessary to paint twenty-four models on twenty-four canvases in twenty hours?

Cheers To Statistics Puzzle Solution


Three artist would do the trick. This is because twenty-four artists would paint twenty-four models in two and a half hours. Since the available time increases eight-fold (2.5 * 8 = 20), it is possible to reduce the number of painters by the same number of times (24 / 8 = 3).

Saturday, October 11, 2014

Flying Car Prototype

In Slovakia local engineers and  designers presented the new prototype Aeromobile  (flying car). An impressive projects representing a - two-seater roadster that can reach 160 km/h on the road more than and about 200 km/h in the air. Under the hood we may find an aircraft engine Rotax 912 with a capacity of 100 hp.  A single tank (aircar runs on regular fuel) is enough for 700 kilometers of flight or fort a trip of 875 kilometers. This original automobile unit is extremely lightweight - it weighs only 450 pounds.





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.

Sunday, October 5, 2014

Faulty Batches

"This time," said the Treasury Minister, "I ditched those dodgy Europeans, and I have assigned the manufacture of our gold coins to five American companies. Look, they are all shining and beautiful, and they are all exactly the same!"

The secretary looked at the coins, weighed some of them, and cleared his throat. "Ahem, Sir. I would like to point out that here we have at least three different kinds of coin; they all look the same, but their weight is different. Would you please come close to the scale? This coin weights 10 grams, as it should, but this other one is 11 grams, while this one is only 9 grams. Obviously two of our manufacturing companies haven't done a good job."

Sad as he could have been, for having been tricked agin by other dodgy companies, the Minister managed to raise his head. "Well.. it's just a matter of finding the fauly ones using the trick that you've showed me, by using the scale only once..."

"Sir. Actually, this is a different problem altogether, we need to find two sources of errors, rather than just one. One batch is heavier, another is lighter. The method I used before will not be sufficient this time. But we can nevertheless find the two offending batches by using the scale once."

How did they manage to use the scale only once?



Notes:
  • You may assume that each batch is made of a large amount of coins (thousands, millions, up to you! :)
  • All coins of the same batch weight the same amount.
  • The storyline in this puzzle follows from the story in Faulty Batch. It is however NOT necessary to have previously read/solved that puzzle in order to solve this one, even though it may be preferable.

Faulty Batches Puzzle Solution

They had to weigh 1 coin from the 1st batch, 2 from the 2nd, 4 from the 3rd, 8 from the 4th, and 16 from the 5th one.

If all coins weighed 10 grams as they should, the scale would display 310 grams ((1 + 2 + 4 + 8 + 16) * 10). However, since one batch has 9 grams coins, and another 11 grams coins, then the total weight of this combination of coins will be:
Total Weight Number of
9g coins
Number of
11g coins
311 1 2
313 1 4
317 1 8
325 1 16
312 2 4
316 2 8
324 2 16
314 4 8
322 4 16
318 8 16
309 2 1
307 4 1
303 8 1
295 16 1
308 4 2
304 8 2
296 16 2
306 8 4
298 16 4
302 16 8

After seeing the solution to this puzzle, it is clear that it would be a lot easier to simply use the scales up to 5 times rather than go through all this, but where is the fun in that?