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I am parent to a 4-year-old son who is mathematically precocious. An example of what I mean (since I'm sure guys like Gauss were proving theorems at 4):

  • He multiplies and divides small numbers easily, like knowing that if there are 3 kids and 12 Halloween treats then everyone gets 4
  • He understands the idea of prime numbers as those numbers of candies which can't be equally divided among his cousins
  • He has figured out by himself the prime numbers up to ~20
  • He is a LEGO fanatic and uses math as necessary - he can calculate that he needs nine 2x4 LEGO bricks to cover a 6x12 plate before he starts.
  • He asked me what 12 times 12 is, then he announced that the carpet in the doctor's office has 144 squares.

I'm wondering how to help him engage more with math since he seems to enjoy it and is good at it. He's not ready yet for "Martin Gardner junior" style questions like "How many balance weighings would it take to identify the one low-weight gem among 9?" And I'm not sure I want to start him on things like memorizing addition and multiplication tables since that's not very relatable to the real world.

What kinds of concepts, questions, techniques, books, activities, etc. should I be introducing him to?

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    $\begingroup$ Fractals are a fun one that may be appealing and have many simple-rules examples. Finding the pattern of the differences between consecutive squares, cubes, etc. is where I started out, but I was a bit older... $\endgroup$
    – abiessu
    Jul 13 at 4:17
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    $\begingroup$ “Arithmetic for entertainment” by Yakov Perelman can be a very good book to study together with your son, or, perhaps even better, look into other books by Yakov Perelman translated into English (do not mistake him with Grigory Perelman who solved the Poincaré conjecture) $\endgroup$ Jul 13 at 4:24
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    $\begingroup$ Another Russian recommendation would be Zvonkin's "Math from Three to Seven". "This book is a captivating account of a professional mathematician's experiences conducting a math circle for preschoolers in his apartment in Moscow in the 1980s." $\endgroup$ Jul 13 at 4:33
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    $\begingroup$ Perhaps pencil-and-paper "logic puzzles", especially those types requiring step-by-step "proofs" rather than guesswork, such as sudoku, killer sudoku, nurikabe, masyu, hitori, heyawake, slitherlink, fences, walls, bridges (hashi, hashiwokakero), fillomino, linesweeper, tapa, yajilin, rooms, cave, tapa, kropki, kuromasu, calcudoku, rectangles (shikaku), area mazes (menseki meiro), train tracks, battleships, star battle, etc., etc. I don't know any specific sources of good puzzles of this sort for young children, but, as suggested in another comment, Mathematics Educators SE may be able to help. $\endgroup$ Jul 13 at 13:21
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    $\begingroup$ Apparently as a young kid, I enjoyed playing with Cuisenaire rods [although I don't know what age I was]: en.wikipedia.org/wiki/Cuisenaire_rods Could be an idea? $\endgroup$ Jul 16 at 10:56
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Here are some random toys, games, and things to make:

https://en.wikipedia.org/wiki/Flexagon#Hexaflexagons

https://en.wikipedia.org/wiki/Sprouts_(game)

https://en.wikipedia.org/wiki/Nim -- It's not that exciting as a game, but it's susceptible to solving by recursion, which is kind of cool. Martin Gardner wrote an article on a version called Wythoff's Nim, which is isomorphic to a game on a chess board. The solution on the chess board is easy to visualize.

https://en.wikipedia.org/wiki/Racetrack_(game)

https://en.wikipedia.org/wiki/Penrose_tiling

https://en.wikipedia.org/wiki/Ultimate_tic-tac-toe

If you don't mind risking having him become a physicist rather than a mathematician, you could encourage him to branch out a little bit from legos and build other types of things. I remember building stuff like a paddlewheel boat that ran on a rubber band, a home-made parachute so I could drop toy soldiers out the window of our third-floor apartment. There are books on origami and paper airplanes.

There are children's books on things like codes and ciphers, Egyptian hieroglyphs.

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Some additional ideas and links:

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    $\begingroup$ I especially like the Chutes and Ladders suggestion, thanks! He is a big fan of Turing Tumble. $\endgroup$
    – Akdinv
    Jul 24 at 0:05

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