## Binomino (Cooperation Games V)

Here is yet another domino variation. Below are the eight 3-binominoes:

You place them sideways so that colors of adjacent squares match. This is, alas, impossible, unless you use two identical binominoes. Therefore we are generous and allow the binominoes to be shifted up or down by one square, so that they have contact only along two squares. A chain using all eight might look like this:

The connectivity graph is surprisingly complicated. I have drawn the directed version, the target of an edge connects to the source by sliding it down one square.

A simple game for two players divides the eight binominoes in two sets of four. Each player gets one set, and they take turns placing binominoes in a chain so they match along two squares. If a player can’t play (either because their binominoes don’t fit, or because they are out of them), they have to take one from either end of the chain. The goal is for both players to finish simultaneously.

Above is a circular version with 4-binominoes, ragged so that not only the colors have to match, but the notches as well. I hope this looks appealing. The same rules apply, but now the goal is to create a closed chain, like so:

Here this is even done so that the binominoes are always shifted the same way. In other words, this solution represents a Hamiltonian cycle in the directed connectivity graph. I believe this graph is always Hamiltonian, for n-binominoes with arbitrary n.

## Recursion (Solitaire XXV)

Time for another Isolation Puzzle while the days grow darker. Here are all 30 squares whose sides are colored in four different colors, chosen from five available colors. Print and cut them all out:

Pick one of them. The first easy puzzle is to assemble it using four of the remaining 29 squares, like so, for instance:

Notice that the border of the 2×2 square matches the border of the chosen square, and squares that touch must match along their common sides, too. The second not quite so easy puzzle is actually four puzzles, namely to assemble each of the four squares on the right using 16 of the remaining 25 squares. No duplicate usage is allowed! Below is a solution for the top left square.

Do this with the other three squares. The solution is by no means unique, and it is well possible that you get stuck and have to revise your choices. When done, you are left with 9 remaining squares. The final not at all easy puzzle consists of assembling them into a 3×3 square so that the sides are in one color each, and again tiles match their colors along joint sides. Like so:

In the above example (yours might well be very different), can you also have dark green occur on the sides of the 3×3 square? And, has this anything to do with dark green missing in the square we started with? Sometimes the past is going to haunt us…

And finally (but maybe this is impossible), can you make the 3×3 square so that its border is an enlarged version of the square we started with?

## Pillowminoes (From the Pillowbook VI)

The admission of an abundance of pillows with straight edges to the zoo raises the question whether these new citizens are any good. We have employed the ones with two straight edges to form arrows and combine to Hamiltonian circuits. Today we will look at those eight pillows with a single straight edge (let’s call them the singles)

Can we use them to tile a curvy 7×7 square, say? The answer is clearly no, because the singles have to hide their straight edges by combining in pairs to pillowminoes. This means we can only tile curvy shapes with an even number of singles. Here, for instance, is a simple solution that shows how to tile a 7×7 curvy square with a gap at the center. It also shows how to tile this shape with a single pillowmino.

This looks too easy? Can we also tile the same curvy 7×7 square so that the missing square at the center has four straight edges?

A little trial and error shows that this is not possible, but we would like to have a reason for this. We need for singles to neighbor the missing square at the center with their straight edges. I have indicated their position by a slightly darker shade of green. Thus the remaining lighter green squares will be entirely curvy, so needs to be tiled with singles that have combined into pillowminoes. That, however, is impossible: Color the squares alternatingly yellow and pink, as in the solution above. Each pillowmino will cover a pink and a yellow square, but the light green shape that needs to be tiled consists 24 pink and 20 yellow squares. This argument also shows that the missing square needs to have all edges curved.

What else can we do with the pillowminoes? There are 36 of them, and not all of them tile by themselves. If we want to tile the curvy 7×7 square with a circular gap in the middle, we will also need to balance the convex and concave edges, as explained earlier.

Here are the ten balanced pillowminoes:

Surprisingly, only the top left will tile the 7×7 (or any larger) curvy square with a central gap.
Below is an example that tiles with four individually unbalanced but centrally symmetric pillowminoes.

More to follow!

## Stars and Stripes

A while ago I had the idea for a card game where each card is a square representing both halves of a domino piece simultaneously. That is, each card is decorated with one to six symbols of one kind (say stars), and one to six symbols of another kind (say stripes). Cards could be placed next to each other if they followed the matching rule that requires them to have the same number of stripes or stars. Here is a chain of eight cards, all following the matching rule:

I liked the idea (and I am sure others must have had it before me), but I also had a hard time coming up with a game worthy of this set of cards. Recent developments triggered the much needed idea.

Stars and Stripes is played on a 7 x 7 board. In the initial setup, the 36 cards are shuffled. Each of the 2-4 players draws a card randomly and places it face up into the corner nearest to him or her to mark home. The middle square of the board is occupied by a special card, called the Trump. You can make and decorate it yourself as you see fit, I have kept it gray and empty. Here is how the setup could look like with four players.

The remaining cards are dealt out to all players. If you play with 2, 3 or 4 players, each gets 17, 11 or 8 cards.

The primary purpose of the game is to gather the largest number of followers. A follower is a card on the board that is connected to the player’s home corner through other cards, who are then followers as well.

The players take turns. At each turn, the player must perform exactly one of the following three actions:

• Place a card from the hand on an empty square of the board that is surrounded in all 9 directions by empty squares or by the border of the board. This card is then called an independent. Placing a card like this can be used to prevent other players to expand their fellowship, or to prepare one’s own future expansion.
• Place a card from the hand on an empty square of the board so that it borders one or more follower cards of the player, but to no follower card of another player. Cards may border independent cards and/or possibly the Trump.
Cards must be placed following the matching rule for all neighbors with which they share an edge. This means that the placed card must have the same number of stars or the same number of stripes as each neighbor in the four directions north, east, west, or south. The matching rule is not applied for the Trump. A card placed this way will automatically become a follower of the player, as do all independents this card possibly connects to.
• Exchange one card randomly with another player. This is done as follows: Both players spread out their cards face down, and both players select simultaneously a card from the other player.

Let’s look at an example. After a few turns, the board might look like this:

Player NW (upper left corner) has five followers, NE and SE four, and SW five. There are three independents. NE is blocked by an independent with one star and four stripes. The only way out of it is to play a card with four stripes and six stars or a card with three stripes and one star. This would also convert the independent into a follower.

Let’s suppose it is SW’s turn, and he or she would like to play the card that I placed next to the board. There are only three possible spots left for this card, marked by roman numerals.

By playing in spot I, SW will gain the independent with two stars and stripes as a follower. Playing in spot II just adds the card as a follower, and playing in spot III connects to the Trump.

Only one player can connect with the Trump, and the Trump does not count as a follower. However, by being connected to the Trump, the player is now allowed to break the matching rule:
Whenever he or she wants to place a new card, this card still must be either isolated or only border the player’s own followers and possibly independents, but the card does not need to match in the number of stars or stripes. In other words, the player connected to the Trump has it much easier to increase the number of followers.

The game ends when after a round, a player has run out of cards or no new card has been placed during that round. The winner is the player with the most followers.

For increased fun, this game can be played also on larger boards with several decks of cards.

You can download a pdf file with cards to print and cut out here.
Get it now, while the game is still legal to play.

## Domino meets Towers of Hanoi

When a neighbor and colleague of mine told me he has a blog about abstract comics, that concept fascinated me to the extent that I had to make one myself. Here it is:

This, by the way, makes a nice poster. I called it Migration, and didn’t give a clue where it came from. There are very smart people who have figured it out by just looking at it, but you can’t compete, because you have already read the title of this post.

Let’s begin with the Towers of Hanoi. This puzzle is so famous that I will not explain it here, mainly because I was traumatized as a high school student when I was forced to solve the puzzle with four disks on TV, in the German TV series Die sechs Siebeng’scheiten. I just pray that no recording has survived.

In any case, after a healthy dose of abstraction, let’s look at the Towers of Hanoi from above, and treat it as a card game.
The disks are replaced with cards that have a disk symbol on it. For the three disk game, there are three different cards, showing a small, medium, or large disk. To make everything visually more appealing, we color the disks, and to emphasize size, we show empty annuli around the smaller disks, as above. Then the solution of the three disk puzzle would look like this:

Because a card hides what is possibly underneath, a position requires context. This is one of the two ways the puzzle is mutating into a story. In the next step, we use domino shaped cards consisting of two squares instead of square cards. Here are the six hanoiomino cards:

The puzzle is played on a 2 x 3 rectangle, with all six cards stacked like this in the top row:

Note that we have modified the Hanoi-rule: In the original version, a card can only be placed on an empty field or on a card with a larger disk. A hanoiomino must be placed so that each of the squares either covers an empty square or a square with a disk of at larger or equal size. This allows for more choice, which causes the second mutation of puzzle into story.

The migration story now tells how to move all the hanoiominos to the bottom row, to the same position, albeit reversed. It is the shortest solution, and unique as such, unless you want to count the backwards migration as a second solution.