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Can you slice a hexagon from a cube?

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Can you make a single cut to a cube to make a regular hexagon? If so, how? And what length are the sides of the hexagon if the cube has sides length 1?

Click to reveal the answer

Well, perhaps unsurprisingly the answer is, yes you can. Cut through the centre points of the sides like below.

If the cube’s side is length 1, then the hexagon’s sides are the longest side of right angle triangles with side length 1/2. So using Pythagorus, its length is the square root of 2 x (1/4), or √2/2.

How’s this possible?

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Why does a one-square hole open up when we rearrange these shapes, as shown below?

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The longest sides of what appear to be “triangles” are bent. You can read a more thorough explanation here.

I’ll try and come up with a more satisfactory explanation.

Divide your timer

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Can you draw two straight lines to cut the clock dial into two sets of numbers with the same sum? And can you do it with five lines to create six groups with an equal sum?

Reveal the solution
With two lines you’re going to get three sections if your lines do not cross. So, to be equal, they must be a third of the sum of the digits on the clock, meaning they are 78/3. Look for groups of numbers lying together with this sum and you will get the top and bottom groups. The middle one will come automatically. Follow a similar process with the five lines. A possible snarl-up would be if you started investigating crossing lines. With the first you could rule it out because 78 does not divide by four. With the second it is in the question. Sorry if this happened. Try to explore the “easy” case first.

Can you halve yin-yang?

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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?

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.

Can you identify a liar?

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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.