Crafty, Math Tricks, Murderous Maths, Together with sis

A Fun Fibonacci Puzzle

Here is a fun math trick! Simon and Neva have made a 8 x 8 cm square (with an area of 64 cm²) and cut it into four pieces, turning the square into a puzzle. Using the same four pieces, they built a 5 x 13 cm rectangle. But wait a minute! 5 x 13 equals 65, so the area of the rectangle is one cm² larger than that of the square!

They also made a similar puzzle using bigger pieces. A 13 x 13 = 169 cm² square turned into a 8 x 21 = 168 cm² rectangle! So now the area of the rectangle is one cm² smaller than that of the square! What’s gong on?

You have probably recognized the numbers in this trick: 5, 8, 13, 21… Those are Fibonacci numbers! Simon explains, that with Fibonacci numbers, the effect of the rectangle area being greater or smaller than the square area is alternating. Fibonacci have a converging ration to φ (Phi), but not φ. The pieces only look like they are golden ratio bigger/ smaller. In reality, there is a little gap between the pieces in the first rectangle and a little overlap in the second.

Simon has been inspired by Mathologer to build this.

the 8 x 8 square
the 5 x 13 rectangle
the 13 x 13 square
the 8 x 21 rectangle
Math Tricks, Milestones, Murderous Maths, Simon teaching, Simon's sketch book

The Math Behind 2048

Simon shares his strategy to win a 2048 game. He has also worked out a general formula of what a maximum tile can be in any grid. For a 4 x 4 grid classic 2048 grid that maximum is 2^17 or 131072!

“It’s a lovely coincidence that there are 17 particles known in the Standard Model of particle physics, and 2^17 is also the maximum value tile in 2048. And so LHC 2048 actually exists!” Simon shouted after we had finished filming. Ten minutes later, walking outside, he calculated that when playing simplest version of 2048, the game of 4 on a 2 x 2 grid, the probability of winning (getting 4) is 19% if you do nothing, 54% if you make one move and 27 % if you make two moves. He also proved that in the game of 4, you win with the maximum of two moves.

2048 offers a lot of opportunities for math fun!

Math Tricks, Milestones, Murderous Maths, Notes on everyday life, Together with sis, Trips

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!

Math Riddles, Math Tricks, Murderous Maths, Museum Time, Simon teaching, Simon's sketch book

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.

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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) :

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Crafty, Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

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 Tricks, Murderous Maths, Simon teaching, Simon's sketch book, Trips

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: