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This yin-yang symbol is made of a circle with two semicircles in the middle to define the border between yin (black) and yang (white) areas. Can you draw a single straight line to divide both areas in half?
This is a well known one, but it came to my attention again thanks to mindbending maestro Martin Gardiner.
Meet Andy, Bob and Chris. We know one of them always tells the truth, one always lies and another does a bit of both. Andy says, “Chris is the one who mixes truth and lies.” Bernie says, “Andy is a liar.” And Chris says, “I mix truth and lies.” Which is which?
Chris can’t be the consistent truth teller, because his statement would be a lie. Now if we imagine Bernie was the truth teller then his statement would mean Andy is the compulsive liar, but he can’t be because Andy’s statement about Chris would be true. So that only leave the option of Andy being the truth-teller, making Chris the one who mixes truth and lies, and Bob the liar.
Adapted from a puzzle by Raymond Smullyan.
Can you connect A to A’, B to B’. C to C’ with no lines crossing.
Taken from a lecture series on topology and geometry by Dr Tadashi Tokieda of Stanford University.
You are in a country surrounded by water and/or impenetrable walls. There are two types of people there, those who always tell the truth and those who always lie. You start walking to the next village when the road splits into two branches. There are two local people standing there. Can you ask a single yes-no question to find out which way to go?
Yes, you can. You can point to one of the branches and ask each person, “Would you say this was the right way?” The non-liar would tell you reliably. The consistent liar, meanwhile, would have to lie in their report about what they would say, so reversing the lie, so telling the truth.
My own solution was clumsier. I’d pointing to one of the branches and ask whether someone from the other group would say yes or no when asked if it was the right way? The liar would have to continue to give the wrong answers, while the truth teller would accurately mimic a liar and give the wrong answer too. So you then just have to do opposite of what they both say.
It is an extension of a similar puzzle with just one liar or non liar, where you can reverse a perpetual liar’s answer by asking, “What would you say if I asked you if you were a liar?” The honest person would say no. But the liar would have to say the opposite of what they would say if asked directly, so telling the truth.
The modelling of real-world liars is simplistic. Real liars are unlikely to feel the need to lie in reporting their own hyperthetical answers, instead they are more likely to lie just as if they had been asked the direct question. Real world liars also have the options of dodging clear questions, by not answering, by giving confusing answers or by giving a clear answer to a different question.
An enterprising liar might ask, “That is an interesting question, but it oversimplies the situation. You might also want to considered the possibility that it might be in the direction you just came from?” Liars don’t lie consistently. Anyone wishing to win people’s trust needs to do so by sticking to a broadly factual basis, expanding this trust on warping accounts in areas which specifically serve them.
Q: You have two long planks between which is trapped a load of gunge with a length of one unit (pictured). You then move one plank down one unit and then push it back up again. You repeat this six times. How long is the gunge now?
Moving the plank up from the position shown on the right it will smear the gunge two units up after the first move. And the same thing will happen next time, moving it three units up, and so on. So after six repetitions it will reach seven units up.
“What doesn’t kill you makes you stronger,” would be among German philosopher Friedrick Wilhelm Nietzsche’s greatest philosophical hits, but researchers scouring some of his late, unpublished writings say he was working on a follow-up saying.
Notes unearthed from towards the end of his life have revealed that he was having second thoughts, and toying with a number of corollaries to the famous saying, in an unpublished volume of notes called Entschuldigung, auf Nachdenken (Sorry, On Second Thoughts).
Among his amendments to a life’s work was, “What does kill you doesn’t make you stronger.” This is thought to have been the result of ruminating on the likely effects of having tertiary syphilis, a condition that is widely blamed for Nietzsche slipping into madness.
“Later in his life, before he lost his mind to dementia, Nietsche seems to have realised that there may have been different qualities and types of suffering, some of them not beneficial to the sufferer,” says Professor Roger Lithcoe, from the Institute of Nietzsche studies at Heissenbaden.
“He was also toying with a follow up to the famous phrase, but sadly died before they were published.” They never appeared posthumously either, “A series of publishers looked at them but didn’t think they were as catchy as his earlier one.”
“It is a great shame that Nietzsche’s On Second Thoughts did not come into the light sooner, or his earlier saying may not have been used so widely to justify suffering sadism, coercion and neglect, often within educational institutions.”
“We only need to look at ideas like, ‘No pain, no gain’, ‘Tough love’, and the idea that suffering build character to see its ongoing influence,” says Lithcoe. “Nietzsche realised suffering is not necessarily of educational value. Endurance has its uses, but it only teaches us so much.”
In a lucid spell just before his death Nietzscheis said to have commented to a nurse, “Ich habe seit vielen Jahren Syphilis, aber ich habe nichts gelernt,” or, “I have had syphilis for many years, but I have learn nothing from it.”