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Japanese ladder games

Have you ever heard of Japanese ladder games? I hadn't until I read an article in June's MAA Focus magazine. Here's a piece of the article:

The MAA Focus article gives a link to a page with a couple of very interesting papers.

The second paper, "Ladders, Codes, and Actions," has this as a first paragraph.

Combinatorial games often generate interesting mathematics and can help students understand difficult concepts. In this paper, we shall describe three games which are very visual and can be played by anyone regardless of their level of mathematical sophistication. When you are done playing these highly addictive games you will have a deeper understanding of permutations and group actions and you will learn a very interesting connection to coding theory. We have found that the games are very enjoyable to play and that players end up understanding complex mathematical ideas without realizing they have done so.

These games are truly a great find, very simple yet very deep. I'll use these games in my Math gatherings.

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  1. Top recommendation for this puzzle! Truly beautiful! I have so many questions before even solving my first one. What permutation of the upper numbers requires the most rungs? How does this increase with N? What about a spider web (cyclic) version?

  2. Interesting. It becomes a game of organized swaps.

    I wonder what the upper bound on number of rungs given the number of inputs is. I can’t seem to find one requiring more than 6 for 4 inputs.

  3. We used to use these as children’s games in school when I was in Korea. They are easy to make and fun for kids.

  4. This is called “Amidakuji” in Japan. Everybody knows this game.

  5. They’re sorting networks (http://en.wikipedia.org/wiki/Sorting_networks) with the additional constraint that only adjacent swaps are allowed. I’m not sure if that’s a common constraint in these puzzles or not?

  6. Ah. From the Wikipedia article on Amidakuji (thanks Shinobu!): “A Ghost Leg can be constructed arbitrarily, but such a Ghost Leg is not necessarily prime. It can be proven that only those Ghost Legs constructed by bubble sort contains the least number of legs, and hence is prime. This is equivalent to saying that bubble sort performs the minimum number of adjacent exchanges to sort a sequence.”

  7. This is the first time I heard about this game but it looks cool and I think my kids will love this game.

  8. We used to use these as children’s games in school when I was in Korea

  9. This ladder game is very famous in Japan. we need to think a lot, While playing this game.

  10. You can read all about the mathematics behind the games at:


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