## 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.

## Monday Math Madness #11

MMM #11 is a variation on MMM #9. I promise I won't do any more variations on this problem after this one!

Consider all of the 6-digit numbers that one can construct using each of the digits between 1 and 6 inclusively exactly one time each. 123456 is such a number as is 346125. 112345 is not such a number since 1 is repeated and 6 is not used.

How many of these 6-digit numbers are divisible by11?

While you may use a computer program to verify your answer, show how to solve the problem without use of a computer.

MMM #9 was interested in divisibility by 8. This contest is interested in divisibility by 11.

I have a Rubik’s Revolution, courtesy of Techno Source, (or $10 Amazon.com gift certificate) to give to the winner. I’ll give more than one prize if I get lots of correct submissions.

I've changed rule #9 to encourage original solutions, which I'm much more likely to acknowledge:

I

maypost names and website/blog links for people submitting timely correct well-explained solutions. I'm more likely to post your name if your solution is unique.

Here are the rules for the contest:

1. Email your answers with solutions to mondaymathmadness at gmail dot com.

2. Only one entry per person.

3. Each person may only win one prize per 12 month period. But, do submit your solutions even if you are not eligible.

3. Your answer must be explained. You must show your work! Wild About Math! and Blinkdagger will be the final judges on whether an answer was properly explained or not.

4. The deadline to submit answers is Tuesday, July 29, 2008 12:01AM, Pacific Time. (That’s Tuesday morning, not Tuesday night.) Do a Google search for "time California" to know what the current Pacific Time is.)

5. The winner will be chosen randomly from all timely well-explained and correct submissions, using a random number generator.

6. The winner will be announced Friday, August 1, 2008.

7. The winner (or winners) will receive a Rubik’s Revolution or a $10 gift certificate to Amazon.com. For those of you who don't want a prize I'll donate $10 to your favorite charity.

8. Comments for this post should only be used to clarify the problem. Please do not discuss ANY potential solutions.

9. I *may* post names and website/blog links for people submitting timely correct well-explained solutions. I'm more likely to post your name if your solution is unique.

## Monday Math Madness #9

Last Friday Blinkdagger announced a winner for MMM #8. Here's MMM #9:

Consider all of the 6-digit numbers that one can construct using each of the digits between 1 and 6 inclusively exactly one time each. 123456 is such a number as is 346125. 112345 is not such a number since 1 is repeated and 6 is not used.How many of these 6-digit numbers are divisible by 8?

While you may use a computer program to verify your answer, show how to solve the problem without use of a computer.

## Monday Math Madness contest #3 ending soon

If you've not gotten your entry in for contest #3 you've only got through Monday night to do so. We've only gotten 10 correct submissions for this contest so far, so your chances of winning the prize are better than they were for the last contest.

## It’s Monday Math Madness time! (contest #3)

We've completed two Monday Math Madness contests. Last Friday Blinkdagger announced that Joshua Zucker, director at Julia Robinson Mathematics Festival, was randomly selected as the winner of the 2nd contest. Now it's my turn to post a contest problem. Those of you who are astute readers may have noticed that I said the contest would be held the 1st and 3rd Mondays of the month and today is actually the 5th Monday of March. Well, there's enough enthusiasm about this contest so we'll just do it every other week. So, we'll do 26 contests per year rather than 24. We're nice that way!

## Mathematician David Gale leaves legacy

Earlier this month UC Berkeley professor emeritus of mathematics David Gale passed away. Gale made a number of significant contributions to mathematics and he loved puzzles, games, and finding beauty in mathematics. Gale's daughter had this to say:

## A very clever way to solve the first Monday Math Madness problem

On March 3rd Blinkdagger and I posted the first Monday Math Madness problem. On March 11th, after the first contest ended, I posted a couple of different solutions to the problem. Pat Ballew, even though he wasn't picked as the random winner, impressed me with a very clever solution to the problem that generalizes very nicely. He uses an approach called Markov state matrices, which I had never heard of. It seems to me that this approach is pretty similar to the one I posted from Richard Berlin. Pat and I exchanged several emails where he explained the method and here is my attempt to explain what Pat explained to me.

This was the problem:

A popular blog has just three categories: brilliant, insightful, and clever. Every blog post belongs to exactly one of the three categories and the category for each post is selected at random. What is the probability of reading at least one post from each category if a reader reads exactly five posts?

Pat's approach starts by creating a matrix that encodes the probabilities of going from one "state" to another as a new blog post is read. State just refers to whether 0, 1, 2, or 3 categories have been encountered after reading some number of blog posts. After one blog post has been read we are in state 1 (1 category has been read). After two posts have been read we may be in state 1 (if both blog posts are in the same category), or state 2 (if the two categories are different), but not in state 3 (you could not have encountered three categories after having read only two blog posts.)

## Monday Math Madness #2 now at Blinkdagger

Blinkdagger has posted the second Monday Math Madness contest. It has a fun St. Patrick's Day theme. Check it out!

You've got a week to solve this problem and send in your well-explained solution. The Blinkdagger guys are giving out $10 in Amazon gift certificate cash to a randomly-selected winner.

April 7th will be the next contest at Wild About Math! I'll be giving away something more fun that cash next time so check back here but solve the Blinkdagger St. Patty's Day problem first.

## Keep those Monday Math Madness answers coming!

Four of you have submitted answers to Monday Math Madness so far but only two of them are correct. Remember, the Blinkdagger guys and I are picking a good **random** answer, not the first good answer. So, you've all still got a chance to win $10 to Amazon.

Heather, from the 360 blog, gave the problem to one of her classes to solve, and told them how to submit their answers. So, if you've got Math students who have learned some probability then have them try their hand at this problem. It's not super difficult but it's tricky enough that only half the solutions submitted were correct.

You all have until Sunday night to get your submissions in. And, if you don't like this problem or miss the deadline, Blinkdagger will be running the next contest, starting in 12 days.

Happy problem solving!

## Eleusis: A different kind of card game

In 1956 Robert Abbott invented a game he called Eleusis. In 1959 Martin Gardner popularized the game in his Scientific American column. Eleusis is what's called an "induction game" which means that players need to try to *induce* the rules of the game. Inductive games are compared to deductive games, where players know what the rules are and try to *deduce* their moves. Yes, I find the terms confusing but, nonetheless, they're worth knowing because if you find yourself enjoying Eleusis you'll know how to refer to that type of game.

Eleusis is played with an ordinary deck of cards. The idea of the game is that one player creates a secret rule and other players have to play cards that satisfy that rule. Ultimately, of course, the goal is to determine what the rule is.