You are journeying through a forest when you come across a fork in the road. One path will take you out of the forest, while the other will take you deeper into the woods to a deadly swamp.
At the fork, there are two guards with a peculiar sign in front of them. The sign reads:
“One of us always tells the truth, one of us always lies. You may ask exactly one question to just one of us. Choose wisely.”
The two guards look exactly the same and there is no way for you to tell which guard speaks truth and which lies.
What can you ask either guard to take the right path out of the forest?
An example of a word game is the concept of kangaroo words. Kangaroos are famous for carrying their babies (joeys) in their pouch. Similarly, a kangaroo word contains another word within itself that is a synonym (a word meaning the same thing). The joey word can be whole (such as [sign]al, where “signal” and “sign” are synonyms), or more typically (and interestingly), it can be split, such as in [ma]scu[l]in[e], where “male” is hidden amongst “masculine”. In this case, the word must be in the right order from left-to-right.
Variations of kangaroo words include anti-kangaroo words – where the word carries an antonym (opposite), such as “animosity” carrying “amity”) – or grand-kangaroo words – where the joey word itself is a kangaroo word, such as “alone” carrying “lone”, which carries “one”.
Try the following puzzle – can you find what the joey word is in each of these kangaroo words?
Sudoku is a mathematic puzzle that has gained considerable popularity in the 21st century, rivalling the classic puzzle that is the crossword. You are given a 9×9 table divided into 9 equal squares, filled with a certain number of digits. Your goal is to fill in the table so that each row, column and subsquare (of 9 small squares) contains every digit from 1 to 9. You are not allowed to have the same number appear on the same row, column or subsquare, as there are not enough spaces for spare digits.
The more digits (“clues”) that you are given at the start of the puzzle, the easier it is to solve it. This begs the question: what is the minimum number of clues that you need to solve a sudoku puzzle?
Sudoku puzzles with 17 clues have been completed traditionally. We know that 7 clues is not enough as the last 2 digits can be interchanged, creating puzzles with more than one solution. Using mathematics, we know that if we can solve a puzzle with n clues, then a puzzle with n+1 clues can be solved as well. Ergo, the answer lies somewhere between 8 and 16.
In 2012, Gary McGuire, Bastian Tugemann and Gilles Civario tackled this problem using one of the oldest tricks in mathematical analysis: brute force. The total number of possible sudoku puzzles that can be generated is 6,670,903,752,021,072,936,960, or 6.67 x 10²¹. After accounting for symmetry arguments (meaning that two puzzles may be essentially identical, but just rotated or flipped), we are left with 5,472,730,538 possible unique solutions.
The team used supercomputers to analyse all of these possibilities to see if any puzzle can be solved with just 16 clues, as the conventional thought was that 17 was the minimum number of clues possible from traditional methods. After a year of calculations, the computer found no sudoku puzzle could be solved with only 16 clues. This was confirmed by another team from Taiwan a year later, proving that the minimum number of clues required for sudoku is indeed 17.
A maze is a puzzle with a simple rule – travel from start to finish. The tricky part is that the path from start to finish is not straightforward, but full of twists and turns. It is usually packed in a compact rectangle or circle, with numerous forks and branches. Because of its simplicity, it has been a popular puzzle for millennia.
Like any puzzle, there are tricks to solving mazes. The most basic, but highly effective rule, is the right-hand rule. This is a form of wall following rule, where you run the maze while tracing the wall your right hand is touching. Keep following a wall with your right hand and you will eventually reach the end.
The rule works for most simple mazes that are simply-connected, where each wall is connected to the outermost wall. But in some cases, the maze is not simply-connected and you will end up in a loop. In this case, you will eventually end back at the beginning, so you will have to try follow a different wall (i.e. use your left hand instead and see how you go).
One of the most famous mazes in history is the Cretan labyrinth, featured in Greek mythology (likely based on the actual palace of Knossos). The Cretan labyrinth was a cryptic maze within the palace the housed the fearsome minotaur, to whom human sacrifices were sent to be devoured. The minotaur is slain by the hero Theseus, who navigates the labyrinth and safely escapes by using a ball of thread given to him by the princess, Ariadne, as a trace.
Mazes were especially popular amongst nobles in Europe, with many castles featuring hedge mazes as part of their magnificent gardens. It is likely that these mazes were popular not because they offer an intellectual challenge, but because it is an ideal date location.
Walking a maze gives you a sense of intimacy, because the paths are narrow and you can only perceive a small space of 10-20 square metres, due to its many-walled nature. You walk side-by-side with each other, while your footsteps echo on the hedges. You have nothing else to distract you other than plain walls and the sky, so you can focus on each other. But most importantly, it provides privacy, by transporting you to a secret, little world of your own.
Imagine that there are twelve coins in front of you. They are exactly the same size and shape, but one is either lighter or heavier than the other 11. To determine which coin is the odd one out, you are allowed to use a scale exactly three times. How do you find the unique coin while figuring out whether it is heavier or lighter than the other coins?
It is recorded that one day, Confucius was presented with a small marble that was filled with tiny holes with twists and turns. He was challenged to try thread the marble with a piece of thread. Confucius tried and tried but could not complete the challenge.
Feeling lost, he asked for some time to think and took a walk. A passing woman noticed him and asked what was wrong. Upon hearing the story, the woman said: “Think quietly, quieten your thoughts”. This gave Confucius an idea, so he thanked the woman and returned to the puzzle. In Chinese, the character for “quiet(密, mi)” sounds the same as “honey(蜜, mi)”. He found an ant, tied a thread around its waist and then placed it on one hole. He smeared some honey on the hole on the other side of the marble and the ant followed the scent, threading the marble as it travelled through the twists and turns.
Thomas Edison famously said that “Genius is 1% inspiration, 99% perspiration”. Although this quote is usually used to stress the importance of effort and striving to succeed, you still need that 1% of creativity and out-of-the-box thinking to achieve true success. The inspiration can be from anywhere – a small memory in the recess of your mind, the casual remark of a passerby, an insignificant detail in your surrounding… What is important that you open your eyes and be open to such inspiration, no matter how silly or lowly you think the source is. You never know what or who will inspire you to have a eureka moment.
Imagine, if you will, a very long piece of ropethat loops around the Earth, fitting it tightly around the equator like a belt. If you wanted to raise this rope off the surface by one metre all around, how much more rope will you need?
The length of rope is the same as the circumference of the Earth which is 40,075km (24,901 miles). Ergo, it is easy to think that you would need kilometres of rope to extend it enough to float a metre off the Earth’s surface. However, in reality you need a mere 6.28m of extra rope to achieve this.
The reason is extremely simple, mathematically speaking. The circumference of any given circle is given by the equation 2πr, where r is the radius of the circle. Therefore, if you increase r by 1 unit (e.g. 1m), then the circumference increases by 2π x 1 = 2π = 6.28. No matter how large the circle may be, this rule does not change.
(This is a famous maths riddle, but here’s a much more interesting application of the concept in this What If? article. God I love that blog! http://what-if.xkcd.com/67/)
You are walking along the road when you reach a fork, where you see two guards chatting with each other. You approach them to ask which way you should go, but you notice a peculiar sign that says: “One of us always tells the truth, one of us always lies”. There is no way for you to know who tells the truth and who tells lies. If you could only ask one question to help you walk down the right path, what would you ask the guards?
The solution is simple: all you have to do is ask “If I asked the other guard which road is the right one, which road would he tell me to take?” then go to the opposite road. The reason being, no matter who you ask – truth-teller or liar – they will tell you the wrong answer. If you asked the truth-teller, he would honestly reply with what the liar would say, which is the wrong answer. If you asked the liar, he would tell you the opposite of what the truth-teller would say, which would ultimately be the wrong answer.
There are 6 matchsticks. Make 4 identical, equilateral triangles.
The hint is that you must think differently to everyone else. If you think like everyone else, you will never find the solution.
There are 6 matchsticks. Make 6 identical, equilateral triangles.
The hint this time is quite the opposite the first puzzle: think like everyone else.
There are 6 matchsticks. Make 8 identical, equilateral triangles.
The hint for this puzzle is that you must reflect on yourself.
A man asked how old a man’s three daughters were. The father replied with the following statement.
“The product of their ages is 36.”
“It’s hard to determine their ages from just that.” the man asking replied.
“The sum of their ages is same as the number of my house.”
“I still can’t figure out the answer!” the man replied again.
“My eldest daughter is blonde.” the father said, and the man, now smiling, replied.
“Oh, is that so? Then I can figure out how old your daughters are.”
How old is each daughter? And how did the man figure it out?
A computer cannot solve this problem, as it can only be solved using human logic.
