philcain

How’s this possible?

15th March 2023

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

Click for answer

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

11th March 2023

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.

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.