Saturday, June 27, 2015

Four Hats

Four men have been buried all the way to the neck, only their heads stick out. They cannot turn their heads, so they can see only in front of them. A wall has been placed between A and B, so that A cannot see the other 3 (B, C, D), and viceversa. All of them know in which position the others have been buried. So, for example, B knows that C and D can see him, even though he can't see them.

A hat has been placed on top of each man's head. All of them know that there are two black hats and two white hats, but no one is told the colour of the hat he's wearing.

four hats
They will all be saved if at least one of them can safely say what colour is the hat he's wearing. Otherwise they'll all be decapitated.

Which one of them saved the day? And, most importantly, how?

Four Hats Puzzle Solution

C saves the day.

D clearly has the most information at his hands, but seeing one white and one black hat doesn't give him any certainty about his own hat's colour. Would B and C both have been wearing the same colour, D would have been able to provide the answer.

But C is one clever guy and he knows that if D doesn't answer, it means that B is wearing a different colour than him. Because B is wearing white, C knows he's wearing black.

Note that A is redundant: the puzzle could have included only B, C, D. That way, the hats would have been three, with two hats of the same unspecified colour, and one other hat of the opposite colour.

Monday, June 22, 2015

Alan and Bert

I told Alan and Bert that I had two different whole numbers in mind, each bigger than 1, but less than 15. I told Alan the product of the two numbers and I told Bert the sum of the two numbers. I explained to both of them what I had done.

Now both these friends are very clever. In fact Bert, who is a bit of a know it all, announced that it was impossible for either of them to work out the two numbers. On hearing that, Alan then worked what the two numbers were!
numbers riddle

What was the sum of the two numbers?

Alan and Bert Puzzle Solution

Starting Numbers Sum Product Product can also be made using
6, 5 11 30 10, 3
7, 4 11 28 14, 2
8, 3 11 24 6, 4 and 12, 2
9, 2 11 18 6, 3 and 9, 2
Bert knows that there are four possibilities for the starting numbers - so it is impossible to work out the starting numbers using the sum alone.

As mentioned in the question, he's quite clever - so he looks at the product that would appear if he used each of the four possible combinations. As shown in the table above, the product that appears can also be made from different numbers.

So, he announces that it is impossible for him or Alan to work out what the original numbers were. He seems to be on fairly safe ground.

Alan is a little bit more devious as well as being clever. Armed with this snippet of information he needs to look for a pair of numbers that give a non-unique sum and also give a non-unique product.
11 is the only non-unique sum which always gives a non-unique product. Alan is very clever and very smug.

Wednesday, June 17, 2015

Weighing an Elephant

elephant riddle
Hundreds of years ago, a king of an Asian country who lived in a port city, received a visit from a king of adjacent country, his friend.

The visitor brought a present with him. It was an elephant.

The visitor gave it and said, "Can you measure the elephant's rough weight in a day?"

The king of the port city consulted with the retainers. "We just have beam scales weighing bags. Do you have any ideas?"

One vassal said, "I can make the relevant measuring equipment assembling large levers and pulleys, your majesty."

"Can you make it in a day?"

"......I can't."

Another vassal said, "How about weighing in pieces after killing the elephant?"

"I won't."

At last they found the method and measured the elephant's approximate weight without sophisticated devices.

What was the method?

Weighing an Elephant Puzzle Solution

Load the elephant onto a boat large enough to carry it. The boat will sink slightly, and you mark the level of the water on the side of the boat. Then you offload the elephant and fill the boat with bags until the boat sinks to the level marked. The bags can be individually weighed using beam scales and the weight of the elephant is the sum of the weight of the bags.

This puzzle is slightly cunning in that it the geographic location of the city is a small clue.

Friday, June 12, 2015

Carpet Layer

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.

Sunday, June 7, 2015

The Farmer's Problem

Farmer John had a problem. There were a group of brigands that had taken all he had... except for three things: his prized wolf, his goat, and a box of cabbages. They were coming after him, to get the rest. These brigands did not like water, so John went to the Blue River, a deep, fast river that no one could swim, and it had no bridges. He always kept a boat there, because he liked to fish, but it was small. So small, in fact, that he and only one of his precious things could be in the boat at the same time.

It sounds simple, right? Ferry one item across at a time, and come back for the others? Well, if John leaves the goat with the cabbages alone on one side of the river the goat will eat the cabbages. If he leaves the wolf and the goat on one side the wolf will eat the goat. If john is there, only he can seperate the wolf from the goat and the goat from the cabbage.

farmer's problem game


How can farmer John keep his possessions safe from the brigands, without losing a single one?

The Farmer's Problem Puzzle Solution

There are two solutions:

Solution A:
1) John takes the goat to the other side, and leaves it there.
2) He then takes the wolf to the other side.
3) He brings the goat back.
4) He takes the cabbages across, leaving them with the wolf.
5) John Comes back for the goat.

Solution B:
1) John takes the goat to the other side, and leaves it there.
2) He then takes the cabbages to the other side.
3) He brings the goat back.
4) He takes the wolf across, leaving it with the cabbages.
5) John Comes back for the goat.

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......!?"