Imagining infinity

Simon: Mom, there’s no number that’s close to infinity.

I: There isn’t, is there? But even when I realise it’s true, I have trouble imagining this.

Simon: This is the way I imagine it: if you collapse an infinite length into some non-zero finite number, then all the numbers should be infinitesimally close together. That is, infinitely close together but not the same.

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Pythagorean triples as a grid

We are watching a Netflix series as Simon comes up to me and says: Mom, give me an odd number! I go, “All right, ahh, 13!” – He starts scribbling something in his sketch book. His Dad and I exchange meaningful glances, we know him too well not to pauze Netflix and wait patiently. Sometimes we try to say quietly, is this going to take long ’cause we are sort of in the middle of a movie here, but we know math goes first. “Look, I have Pythagorean Triples now!” Simon triumphs. “I simply square your number, then divide the result by two and the two numbers around that are the two missing numbers!.. Do you know how I see it? I basically imagine a grid, a 13 by 13 grid. ” (He starts drawing the grid). “Look and then it has 169 cells in it, and you try to divide it in two nearly identical grids, 84 and 85 cells in each… Hey Mom, do you know I can also make scaled Pythagorean Triples? ”

Three chess puzzles

Set the six figures (pawn, queen, rook, bishop, king, knight) on a chessboard in such a way that every marked cell is threatened that number of times. For example, if the cell is marked “2”, it can only be threatened twice and if it’s marked “0”, it should not be threatened at all.

Simon shows how to set the first couple of figures correctly on every pictures below. Can you finish every puzzle?

A quiz full of fun tricks

Simon got these from the Scam School channel.

I first only got a strip paper with a sequence of green sticks written on it, separated by comas. Simon did tell me those were pieces of numbers, the way they appear on a calculator screen, and that I was supposed to comolete the sequence. I tried to give this one a shot. To stop the waiting that began to seem eternal he eventually gave me the answer: the green sticks were everything BUT the numbers! Below you see the competed version:

Here, too, one is supposed to continue the sequence of numbers written in a column. You see the pattern?

Here too, come up with the next character:

See the pattern now?

The next one is hilarious! Rearrange the letters below to write ONE WORD:

And in the last puzzle, the nine words used to have something in common, but now it’s just eight of them.

If you get stuck with any of these, leave a plea for help in the comments!

Recursive Math Tricks

The above formula allows you to take any number (n) and get it back as a result of many calculations. Simon remembered we had read about it (something like a year ago) in a Russian book by Levshin and found rhe trick description again.

The formula below is for doing more or less the same to any seven digit phone number (the number below is random).

A new tour of Simon’s sketch book

The fertility formula, to predict the population the following year:

A fake number (called “Wau”) to imagine infinity (via Numberphile):

Drawing a square root of 5 (via James Grime):

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Pebbling a Chessboard (via Numberphile):

Kolakoski Sequence:

Proof for probabilities in a Wythoff’s game

Probability that everyone will be eliminated simultaneously in Simon’s “Hat Game” (a card game he invented):

Finite List of Primes:

Creating consecutive numbers by using various operators to connect four fours: