False Proofs: Can you figure out what’s wrong?

Simon shows three false proofs. Can you find the mistake in each proof? Simon reveals the answers to the first two. Try to give your answer to the third one. 

And the answer is: 


Simon’s Proof for the Hundred Board Trick

Simon learned a new math trick last night called Hundred Board. He came up with two ways to prove why the trick always works:

Simon turned 9!

– Simon, Mom and Dad arranged it quite nicely, to have your birthdat and our wedding anniversary on two consecutive days!

– No, it was pure coincidence!

– But what was the chance that Mom and Dad’s wedding dat was one day before or after your birthday?

– One in 182.62125 exactly! It’s because in the Gregorian calendar, a year lasts exactly 365.2425 days.

We set up a treasure search with science questions to look for the 9 presents!

Simon’s sis Neva made the e below.

And the leaning tower on Lire (with the top tile not overlapping with the bottom one) was finally a success  with these foam dominos!

The Leaning Tower of Lire

Also known as the Book-Stacking Problem. Simon had tried to build this tower at the Fries Museum where we visited a huge Escher exhibition (to the annoyance of the museum staff, to whom I had to explain that it was a serious math experiment and not just a kid dropping bricks), but it only worked with 4 blocks (possibly because the blocks were made of foam and weren’t rigid enough). He tried to stack the blocks on top of one another, shifting every next block first by one eighth, then by one sixth, next by one fourth, and next by one half – in the end, the top block would no longer be positioned above the bottom block.


He repeated the experiment at home, first doing some calculations and then using more rigid wooden blocks and managed to stack a tower of 6 blocks! (The top block still overlapped the bottom one by a bit though) :





The Pi Strip

Simon made a measuring tool to check the diameter of round objects: by wrapping the strip around them, he reads the Pi times the centimeters value, which basically gives him the diameter (as the circumference equals Pi times the diameter).

And here he is, measuring the diameters of Neva’s and Dad’s necks:

Math on the Beach

Simon doing math everywhere.

And he showed me this beautiful trick of two rows adding up to equal numbers and their squares adding up to equal numbers. And the two rows below? Even their cubes!

Now, can you come up with two rows in which also the fourth powers add up to equal sums?

Simon learned this trick from Matt Parker: you should pick numbers up to n-1, where n is the next power of 2. In this case, n would be 2 to the fifth power and that is 32, so we pick numbers up to 31. Then we write them down in two rows in such a way that the top row only has numbers whose binary expressions have an even number of ones and the bottom row – only odd number of ones.

Simon also came up with an interesting fact about the trick using a pattern of “buckets” turned in opposite directions:

Tricks with paperclips and Knot Theory

Simon is pretty obsessed with Knot Theory at the moment (a mathematical theory that is widely used in advanced biology and chemistry, for example in handling tangled DNA).

He also learned a few tricks from one of his favourite teachers on Numberphile – Tadashi Tokieda – that probably also have something to do with Knot Theory. By folding a strip of paper in a certain way and placing rubber bands and paper clips on it and then pulling the ends of the paper strip, Simon gets the paper clips and the rubber bands linked together:

Making mathematical knots using rubber bands. A trefoil knot (the main prime knot):



Simon says “it’s good for meditation”, too: