### Brain Teaser Wednesday #btw

The human brain is an amazing thing. It is capable of coming up with wonderful ideas and solving incredibly difficult problems. But, like everything, the brain needs maintenance. I think one of the best ways to keep your mind healthy is to exercise it regularly, so some months ago I set up Brain Teaser Wednesday (#btw). A new puzzle appeared weekly from April to December 2016, all the puzzles from the competition are shown here with their solution, and occasional hints. Each puzzle has a difficulty level, marked by the number of stars: easy*, medium**, hard***, and challenging****.

**Christmas Special**

To end #btw there was a 'Christmas Special Challenge', with the winner receiving a custom-made mug. The Challenge had several different parts, posted every few days over the holiday period. It ran from Monday 19th to Saturday 31st, and only one person managed to complete the whole challenge. You can read all the parts of the puzzle and the final solution here.

Christmas Special Part 1:
First think of the person who lives in disguise,
Who deals in secrets and tells naught but lies.
Next, tell me what's always the last thing to mend,
The middle of middle and end of the end?
And finally give me the sound often heard,
During the search for a hard-to-find word.
Now string them together and answer me this,
Which creature would you be unwilling to kiss?

Christmas Special Part 2: (you don't need brute force!)
RCRTYQEVEK ECABTEEK QEVEKTEEK QCXTEEK TWL BUKDOEDQEVEK QEVEK TBCOTYTWL ECABT LKE TWEHVE RLOTYKCKE TWEHVE TBCOTYTWL. RCRTYECABT. ECABTYQEVEK, EHEVEK, ECABT, RCRTYTBOEE ECABTEEK QEVEK. QCXTYQCX RLOTYECABT RLUO TBCOTEEK ECABTYQEVEK RCRTEEK TWEKTYTBOEE EHEVEK EHEVEK, TWEKTYTBOEE TBCOTYKCKE RCRTYECABT RCRTYQCX RLOTYECABT,

Christmas Special Part 3:
49 441 729 529 121 361 16 1089 225 576 4 961 25 1225 676 64 36 484 9 81 289 1156 1 625 841 100 400 256 324 144 1024 169 784 196 900
Part two tells you 'which', part three tells you 'where'. If you seek guidance, find what parts two and three have in common. If you are lost, remember how everything started and how far we've come.

Christmas Special Part 5:
If you correctly combined parts two and three, you should have already found (and solved) part four. For part five, you need to find the next number in each sequence:
1, 1, 2, 3, 5, 8, 13, ??
1, 7, 10, 13, 19, 23, 28, ??
1, 3, 6, 10, 15, 21, 28, 36, ??

Christmas Special Part 6:
You should now have a name (part four) and six digits (part five). Together, they give you access to a hidden place. This is the place you should go, that is where everything ends. If you are lost, remember where everything else is and go home.

Puzzle 35** (14/12/16): After a big family dinner you have 25 empty bottles, so you all decide to play a game to choose who will be crowned 'Puzzle Master'. The 25 bottles are placed in a square at regular distances, forming a 5x5 grid. You also have five circular plastic rings, which you can set to any size you wish. The winner of the game will be the first person who manages to place all five rings on top of the bottles in such a way that every bottle has at least one ring on top of it. Where should you place the rings to receive the title of 'Puzzle Master'?

Puzzle 34** (07/12/16): Anna, Brian, and Candela reach the final of a paint-ball tournament. To decide the winner a Western-style three-way duel is proposed: they will take turns shooting, and when a person is hit they are eliminated and can't shoot any more. Anna will shoot first, followed by Brian, and then Candela, repeating until only one person remains. Candela is a good shot and hits her target every time. Brian isn't so good, and only hits his target two out of three times. Anna is a very bad shot, only hitting her target one third of the time. What strategy should Anna adopt to maximise her chances of winning the tournament?

Puzzle 33*** (30/11/16): On a perfectly circular island there are six villages along the coast, evenly spaced so the distance between any two neighbouring villages is the same. There are straight paths through the island connecting every pair of villages. Every day you must journey to the village furthest away from your home, always obeying the following rules. You must travel in a straight line between villages; you can only change direction at the villages. You can stop at any village or villages on the way, but you may not visit the same village twice on one journey. You may not travel to a village further away from your final destination than your current village. At the end of the day, you will take the shortest path back to your home village. Assuming you always follow the rules, how many days will it take you to cover every possible route between your home and the furthest village?

Puzzle 32** (23/11/16): One day scientists find a strange genetic mutation, referred to as Mutation Z, present in one person out of every 1000. The geneticists design a genetic test that is able to correctly identify everyone who has Mutation Z, but the test also has a false-positive rate of 5 percent, meaning the test will wrongly find Mutation Z in 5 percent of the people who do not have it. If we choose a person at random, without knowing anything else about them, and the test shows they have Mutation Z, what is the probability that this person actually has the mutation?

Puzzle 31* (16/11/16): While enjoying a nice wedding reception, the newly weds decide to give an expensive bottle of wine to one of their guests, so they propose a game. First, all 42 guests (over the drinking age) will stand in a circle. The youngest guest will be given a ball, which they will throw at the person standing to their left. This person sits down, and the ball will move clockwise to the next person standing, who will repeat the process (guest 1 eliminates guest 2, guest 3 eliminates guest 4, and so on). The whole process is repeated until only one person remains standing, who will then be given the bottle of wine. As a good logician, you know the game is fully deterministic and the winner can be calculated in advance. How close to the youngest person should you stand to guarantee you are the last person standing?

Puzzle 30*** (09/11/16): One day you realise you've had enough of your usual life, so you decide to join a monastery with highly-logical monks. In order to be accepted into the monastery you must prove your logic skills, so the monks sit you down next to a square, rotatable table. After blindfolding you, the monks tell you that they have placed four glasses, one at each corner of the square table, with random orientations (facing up or facing down). Your mission is to get all the glasses with the same orientation, either up or down, while obeying the following rules: you may pick up any two glasses and reverse one, both, or neither of them. Once you put the glasses back down, the table will be rotated a random angle, after which you can repeat the process. As soon as you have all four glasses with the same orientation, the monks will remove your blindfold. Without relying on luck, can you find a strategy that will allow you to succeed with only four rotations of the table (giving you five steps)?

Puzzle 29** (02/11/16): Four bandits ride into town on their horses. They leave the horses tethered outside a bank, which they then go and rob. After robbing the bank, the bandits need to make a quick escape. Each bandit has their own horse, but as they are leaving quickly the first bandit to exit the bank randomly takes one of the four horses. When the second bandit comes out of the bank, if his own horse is not yet taken, he will choose his own horse, but if his horse has already been taken, he will randomly choose another horse. The third bandit will do the same. When the forth bandit leaves the bank, what is the probability that the remaining horse is his own?

Puzzle 28** (26/10/16): While organising an international cosmology school, Prof. Forgetful wasted coffee on his notes, erasing most of the information. He knows there will be one lecture per day from Monday to Thursday where they will discuss the CMB, Dark Energy, Dark Matter, and gravitational waves. He knows the speakers are called Albert, Emmy, Lisa, and Nikola, and they come from Heidelberg, London, New York, and Rome, but he doesn't know how the information fits together. Now he needs your help to figure out which lecture takes place on which day, who will be giving each lecture, and where each person comes from. He was able to recover the following details from the notes:
1. The person who will discuss Dark Matter will give a talk one day after Nikola.
2. The speaker from New York will give their talk sometime after the expert in gravitational waves.
3. Neither the person from Heidelberg nor the Dark Matter expert is Albert.
4. The person from Heidelberg will give a talk one day before the CMB expert.
5. Of the person from London and the person scheduled to talk on Wednesday, one will discuss gravitational waves and the other is Nikola.
6. Nikola will give his talk two days before the gravitational waves expert.
7. Lisa won't give a talk on Wednesday.

Puzzle 27***(19/10/16): Recently, a group of astronomers detected a strange periodic radio signal containing a combination of short and long pulses, coming from a nearby galaxy. Denoting the long signals as '1' and the short signals as '0', the signal was “11110001111000”, repeating over and over. After a while, the signal changed to “001100000000000101000000000000110000011”. The astronomers believe it's a message from an intelligent civilisation, and that the first part of the signal tells us how to understand the second part of the message. Knowing that you are good at solving puzzles, the astronomers ask you to help answer three questions. What does the message say? What does it mean? How many fingers do you expect the aliens to have?
Points: 1 point for each question answered correctly, maximum 3 points.

Puzzle 26* (12/10/16): Marie and you are engaged in a battle of wits over a chess board to decide who eats the last brownie. After defeating you, Marie decides to give you a puzzle. If you can answer the puzzle on the first try, she will share the brownie with you. Her puzzle is: “If we count all the squares of different sizes, there are over 200 squares on a chess board. If I place three pawns in three random squares, what is the maximum amount of squares we could find that do not contain a pawn?”

Puzzle 25**** (05/10/16): An evil demon captures 100 people and decides to play a game with them to see if he sets them free or harvests their souls. He assigns each person a different number between 1 and 100, and writes each number on a separate piece of paper. In a separate room he has 100 boxes in a line, and in each box he randomly places one of the numbered pieces of paper. One by one the people enter the room, where they can look in any 50 boxes of their choosing. After opening 50 boxes they have to tell the demon (in private) which box contains the piece of paper with their own number on it. If all of the people manage to find their number, they will all go free, however if any of them fail to find their own number, they will all be killed. The people are allowed to discuss a strategy beforehand, but once the first person has opened a box they can no longer communicate with each other. Which strategy will allow them to survive with more than 30% probability?

Puzzle 24** (28/09/16): A private detective is following a man suspected of having ties to a criminal network. While buying her usual salad box for lunch, she notices someone slip the suspect a note. With an accidental-coffee-spill trick, she is able to acquire the note, where she reads the following text: “MAATDIETGEAGETLLYHTHEMATMEHOTPEEONEM”.
When and where are the suspects meeting?

Puzzle 23** (21/09/16): While out camping in a forest, you are captured by a group of witches. They lead you to a sacrificial altar where a pentagram (five pointed star) has been drawn. The witches give you two straight sticks, and tell you “You are to place these two sticks on the pentagram such that there are ten non-overlapping triangles. If you succeed, you will be allowed to go free. If not, you will be sacrificed.” How should you place the sticks to survive?

Puzzle 22*** (14/09/16): You have two coconuts and access to a 100-floor building. You want to know which is the highest floor from which you can drop a coconut without it breaking. Both coconuts are identical, and if a coconut is dropped and does not break, it is undamaged and can be used again. If the coconut breaks when dropped from floor 'n', it would also break from any floor above that, and if a coconut survives a drop from floor 'n', it would survive the drop from any lower floor as well. What strategy should you adopt to minimise the number of coconut drops it takes to find the solution? With this strategy, what is the worst case for how many drops it will take?

Puzzle 21** (07/09/16): An anthropologist travels to a recently discovered town hidden in a forest, to learn about the town's people and their customs. In the centre of the town she finds three men with the following sign in front of them. “The Three Wise Brothers can answer any question you have, but be warned: the oldest always tells the truth, the youngest always lies, and the middle one will lie or tell the truth depending on the position of the stars and his general mood. You may ask one yes/no question to any of the three brothers, and then you must choose which brother will answer all your remaining questions. The other two brothers will leave. If you pose a paradoxical question, all three brother will leave.” What question should she ask to guarantee she can choose a brother with whom she can communicate?

Puzzle 20* (31/08/16): One day you are abducted by hungry aliens who are trying to understand our rules of geometry. They show you a 4 meter high stone cube that has been painted bright blue. They ask you the following: “If you were to cut the cube into smaller cubes, each one meter high, how many small cubes would have no sides painted blue, how many would have one side painted blue, how many would have two sides painted blue, and how many would have three sides painted blue?” Obviously, you only have one chance to get the correct answer, else the aliens will eat you.

Puzzle 19*** (24/08/16): You are competing in a series of shenanigans, and for your final task you are in a room wearing a blindfold. You are led to a table with 87 coins, and you are told 24 of them have tails facing up, and the rest have heads facing up. You can flip and move as many coins as you like. Your task is to separate the coins into two piles that contain the same amount of tails facing up. You can't remove your blindfold, nor can you distinguish heads or tails by feeling the coins. How can you do it?

Puzzle 18*** (17/08/16): After sailing to a strange island, three sailors are captured by cannibals. The cannibals decide to play a game with the sailors to settle their fate. The three sailors are ordered to stand in a line, such that the one at the back can see the other two sailors, the middle sailor can see the one at the front, and the one at the front can't see any of the others. The cannibals inform them that they have three white hats and two black hats, and they will randomly place one hat on each sailor's head. No sailor can see his own hat, and they are not allowed to communicate in any way. At any point, any of the sailors can state the colour of his own hat. If the sailor is correct, all three of them can go free. If the sailor is wrong, all of them will be eaten. If after one hour none of the sailors have said anything, they will all be eaten. How can the sailors guarantee their survival?

Puzzle 17* (10/08/16): One day you are out having a coffee with two friends. You know Tim always lies on Monday, Tuesday, and Wednesday, and always tells the truth all the other days. Martha always lies on Thursday, Friday, and Saturday, and always tells the truth on all the other days. They make the following statements:
Tim: “Yesterday I was lying.”
Martha: “So was I.”
On which day of the week did this conversation take place?

Puzzle 16** (03/08/16): Three professors and three students are out celebrating the end of exams. They come upon a river they want to cross, but there are no bridges. They have one small row boat that can carry a maximum of two people, but they can use it as many times as they like. All six people know how to row and none of them want to swim. The professors are feeling a bit mean after the exams, and they decide if at any point there is a student outnumbered by professors on either side of the river, all the students will have to take another difficult exam, but the students get to decide how they all cross the river. How can the students get everyone to the other side of the river without having to take another exam?

Puzzle 15**** (27/07/16): Three mathematicians are playing a game. Gary thinks of two natural numbers (a and b), both larger than one, and Sarah and Peter both have to guess the numbers. Gary tells Sarah the sum of the two numbers (a+b) and he tells Peter the product of the two numbers (a*b). He tells them both that b is bigger than a (b$\gt$a) and the sum of the two numbers is smaller than 100 (a+b$\lt$100). The following conversation takes place:
Peter: “I don't know the numbers.”
Peter: “Now I know a and b!”
Sarah: “In that case, I also know a and b!”
Which numbers a and b was Gary thinking of?

Puzzle 14** (20/07/16): You have three pairs of ping pong balls: two blue balls, two white balls, and two red balls. In each coloured pair you know one ball is slightly heavier than the other. All the heavy balls weigh the same, and all the light balls weigh the same. You have a balance scale, but you are only allowed to use it twice. How can you identify the heavier and lighter ball of each colour?
Note: the scale only has three positions: balanced, left side heavier, and right side heavier.

Puzzle 13* (13/07/16): One day you are visiting a strange island and you are captured by cannibals. The cannibals love maths and they frequently play games with numbers. They decide to give you a challenge; if you can solve it, you go free, but if you can't solve it, they'll eat you. Your challenge is to find a 10-digit number, where the first digit is how many ones the number has, the second digit is how many twos the number has, .... etc ... , the ninth digit is how many nines the number has, and the tenth digit is how many zeros the number has. You only have one chance to get the correct number, if you get it wrong the first time, the cannibals will eat you.

Puzzle 12*** (06/07/16): Five pirates find a treasure chest with 100 gold coins, so they decide to split the coins following specific rules: the oldest pirate proposes how to share the coins, and all the pirates (including the proposer) vote for or against the proposal. If the proposal has 50% or more acceptance, the coins are split following the proposal. If the proposal has less than 50% acceptance, the oldest pirate is thrown overboard and killed, and the whole process is repeated with the remaining pirates. The pirates are rational, greedy, and bloodthirsty; they want to maximise their income and if two proposals give them the same income, they will choose the proposal that results in more deaths of their comrades. Assuming all pirates know and follow these rules, what distribution is the oldest pirate going to propose?

Puzzle 11* (29/06/16): You find yourself in a house with four light switches in the basement and four light bulbs in the attic. You know each switch controls one bulb, but you don't know which switch controls which bulb. If you can only go upstairs once, how can you find out which switch controls which bulb?

Puzzle 10** (22/06/16): One day at 8:00 a monk starts to climb a mountain to reach the temple at the top. He follows a narrow path that winds around the mountain. Sometimes the monk walks fast, sometimes he slows down, and he stops often to eat and enjoy the scenery. Finally, he reaches the temple at 21:00.
After a few days of meditation, the monk leaves the temple. He sets off at 8:00 and takes the same narrow path back down, walking at different speeds and enjoying the scenery. He reaches the bottom of the path at 19:00.
Is there any point on the path that the monk occupied at exactly the same time of the day on both trips? Justify your answer.

Puzzle 9*** (15/06/16): You have 3000 apples that you want to transport to a city 1000 km away using your camel. The camel can carry a maximum of 1000 apples at a time. Additionally, the camel needs to eat one apple for every kilometre he walks; he walks one kilometre, and then eats an apple. What is the maximum amount of apples you can transport to the city?

Puzzle 8** (08/06/16): A farmer wants to start an orchard following very specific rules: he wants five straight rows of trees, each of which must contain exactly four trees; no row can have more than four trees. However, the farmer only has ten trees he can use. What pattern or template should he use to plant the trees?
Bonus question: If the rows can have more than four trees, can you find an alternative solution?

Puzzle 7* (01/06/16): You have ten bags, each filled with approximately 100 coins, but you don't know the exact number in each bag. Nine bags contain real coins and the other bag contains counterfeit coins. You know that the real coins weigh 1 gram, while the counterfeit coins weigh 1.1 grams. You have a very precise digital scale, but you are only allowed to use it once. How can you find the bag that contains the counterfeit coins?
Note: you are allowed to remove the coins from the bags, just be sure not to mix them up.

Puzzle 6**** (25/05/16): There is an isolated monastery where highly logical monks live. They have all taken a vow of silence and can't communicate with each other in any way. Furthermore, they are forbidden from seeing their own reflection: there are no reflective surfaces in the monastery. Every morning, all the monks sit down together and have breakfast. The monks have either brown eyes or blue eyes. Blue eyes are considered a curse, and any monk who discovers he has blue eyes must leave the monastery after breakfast. One afternoon a tourist visits and announces to all the monks 'At least one of you has blue eyes'. On the sixth morning after the announcement, after breakfast, all the monks with blue eyes leave the monastery. How many monks had blue eyes, and how did they know they had to leave?

Puzzle 5* (18/05/16): You have two ropes of different lengths and a box of matches. You know it takes exactly one hour for each rope to burn, but the burning rate is completely irregular, meaning you can't say it takes 30 minutes to burn exactly half of the rope. Without using anything else, how can you measure 45 minutes?

Puzzle 4** (11/05/16): Four people are travelling at night and they come upon a narrow bridge that is only big enough for two people at a time to cross. They only have one torch and everyone needs the torch to cross: it's impossible to cross without the torch. Alex can cross the bridge in 1 minute, Beth needs 2 minutes to cross, Charlie takes 5 minutes to cross, and Dany needs 8 minutes. If two people are crossing the bridge, they will only be able to go as fast as the slowest person crossing. What is the minimum time needed for all four travellers to cross the bridge?

Puzzle 3*** (04/05/16): Two mathematicians, Sarah and Mike, meet up after many years. The following conversation takes place.
Mike: 'I have three children. The product of their ages is 72, and the sum of their ages is equal to the number on this door.'
After looking at the number on the door, Sarah comments: 'I still don't know how old they are, what else can you tell me?'
Mike: 'The oldest one likes cheesecake.'
Sarah: 'Now I know how old they are!'
How old are Mike's children?

Puzzle 2** (27/04/16): You have three jugs; one can hold 3 litres, one can hold 5 litres, one can hold 8 litres. The 8-litre jug is full of water, the other two are empty. You have no other measuring devices or jugs. You want to have two jugs with four litres each. How do you do it? (It can't be done by eye or by guesswork.)
Points: 1 point if you need 8 steps or more, 2 points if you need 7 steps, 3 points if less than 7 steps

Puzzle 1** (20/04/15): You have 27 coins, one of which is fake. The real coins all weigh the same, but the fake one weighs a bit more than the others. You have a beam balance scale ⚖. What is the minimum number of weighings you need to find the heavier coin? (Don't rely on luck, the answer should work always).

#### 1 comment:

1. Ahh,the solution to 4 is excellent and obvious now.