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Story

Can you halve yin-yang?

8th March 2023 by philcain

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?

Reveal the solution
We can start by patiently working out the areas of the different elements of the symbol. We find each quarter of the large circle is πr²/4, where r is the radius of ths circle. We also find that the areas of each of the two semicircles is πr²/8, half the area of the quarter segment of the large circle. That is a crucial “coincidence”. And. moving on, πr²/8 is also the area inside a quarter circle on either side of a line drawn at 45 degrees the horizontal (the red dashed line). If we now pause and tot up the areas of white and black each side of this same line we get 2 x πr²/8 each time. So this line cuts each of the coloured area in half. Job done. Some investigation was needed, but spotting a “coincidence” was really what makes this one click into place almost by itself.

This is a well known one, but it came to my attention again thanks to mindbending maestro Martin Gardiner.

Filed Under: Story Tagged With: mindful distractions

Can you identify a liar?

3rd March 2023 by philcain

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?

Reveal the solution

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.

Filed Under: Story Tagged With: mindful distractions

Stretch your thinking

1st March 2023 by philcain

Can you connect A to A’, B to B’. C to C’ with no lines crossing.

Reveal the solution

Taken from a lecture series on topology and geometry by Dr Tadashi Tokieda of Stanford University.

Filed Under: Story Tagged With: mindful distractions

Can you force a liar to tell the truth?

25th February 2023 by philcain

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?

Reveal the answer

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.

Filed Under: Story Tagged With: mindful distractions

Planks and gunge

22nd February 2023 by philcain

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?

Reveal the solution

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.

Filed Under: Story Tagged With: mindful distractions

Unreal: Nietzche’s second thoughts on suffering

17th February 2023 by philcain

“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.”

Filed Under: Story

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