Alexandre Muñiz posted some combinatorial musings on the World Cup recently, and I had a stab at the main posed problem. Figured out some shortcuts, but it's quite combinatorially tricky.

The question is this: given that the soccer teams in each of several groups are evenly matched, and there is a certain likelihood of each game resulting in a tie, how likely is it that the groups will all have different scores?

**( One imperial long butt-ton of explorationCollapse )**

Well, that's about it for the main problem. I don't know the answer. But to make sure my methods work, I tried them in a spreadsheet on three groups of three teams each. The probability of no repeats when ties are 1/3 probability is about .53315 or exactly 1166/2187. This reaches a maximum of about .533468755 when ties have a probability of about .34308.

- Current Music:OK Go, "The Writing's On the Wall"

This is a Star Battle variant. There are fifteen colored letter regions and six white regions. The internal borders in the W, P, A, and E are purely to emphasize the letters.

**Rules:**Forty-two squares of the grid each contain one star, and the rest are empty. No two star squares share a corner or a side. Locate these stars, given that there are two per region, and three per row and column.

So Star Battle is a pretty obvious choice, since Twilight's cutie mark is a group of stars. But why 21 regions in a 14*14 square? Well, I hit upon using 42 stars because in the episode "Fall Weather Friends," some of the cast participated in a race called the Running of the Leaves, and Twilight's racer number was 42. And beyond that number having a prominent place in geek culture, it's also the number of points on the stars in her mark (two large overlapping stars, five smaller stars surrounding them, each with six points).

Then with fifteen regions forming letters, 21 total regions was obvious. And I've already seen several small-region SBs, so using a grid smaller than 21*21 wasn't a tortuous decision.

Next week, I hope to have a doozy of a puzzle ready for you. Maybe I'll even make one for Spike. ^_^

- Current Mood: accomplished
- Current Music:Smash Mouth, "All Star"

This is a kuromasu with the twist that half the clues are multiplicative. Of course, I'm going to de-jargon that for you.

**Rules:**Blacken some of the grid squares so that each number represents its square's line of sight in one of two ways. An observer at each clue looks up, down, left, and right, and sees up to the edge of either the grid or the nearest black square. A clue number is either the count of all these squares including itself, or the total horizontal distance times the total vertical distance. Six of these clues should be one kind, and the other half-dozen should be the other kind.

For instance, the 25 clue on the bottom is obviously the second kind, with five squares visible above and five squares visible left and right. These five-counts include the clue square, so the normal clue that would otherwise occupy that square is 9. (Of course, if a clue can only see its own square horizontally, it gives the same value whichever type it is. Same if it only sees its own square vertically.)

Two final rules govern the black squares. First, no two may touch at an edge, although corners are fine. Second, the black squares must not cut off one section of the grid from the rest.

The highlighted clues that spell Fluttershy's name trace a double-sided spiral, a reference to "Hurricane Fluttershy." I chose kuromasu because Fluttershy has an interesting dynamic with being seen. For the most part, she's not comfortable being the center of attention (see "Green Isn't Your Color" or the aforementioned "Hurricane Fluttershy"). On the other hoof, she has a stare to rival Zoolander's "Blue Steel," powerful enough to resist a cockatrice's magic. That's why I put in two different clue types.

To end on a pleasant note, not only have I composed Twilight's puzzle, but I am ready to follow it up with a seventh going on the "Friendship Is Magic" theme. Well, more or less. You'll see in two weeks (if I don't space things out again).

- Current Music:Smash Mouth, "I Just Wanna See"

Colors, as usual, are courtesy of the Kinky Turtle. The rules are as follows.

Your aim is to shade in a number of half-cell triangles so that the remaining white space forms a number of rectangles, many tilted at 45 degrees to the grid, that share no edge. Rectangles may touch corner-to-corner or corner-to edge. Triangles may touch anywhere. (As a solving aid, each square cell is divided into quarters. If any part of a given cell is shaded, it is exactly two adjacent quarters.)

Each letter represents a distinct digit from 0 to 4, inclusive. This clue number is the count of black triangles that are to share an edge with that clue's cell in the solution. The meaning of each letter is for you to determine.

- Current Mood: okay

This here puzzle's a cryptic variant on Country Road. Whatcha need to do is draw a non-crossing loop from square to edge-adjacent square so's there's one segment in each outlined region.

Two provisos there. First, y'all don't need to visit every square, but where two squares share a thick border, ya gotta visit one or both. An' second, those letters represent numbers, cryptogram-style, an' those numbers are the number of squares in that respective region's segment.

There ya go, third pony puzzle on the LiveJournal. Ah figger ah'll put Rarity's up next week, just fer completeness' sake. Yee-freakin'-haw!

- Current Mood: tired

*exactly*that post.

I haven't exactly had a packed schedule the last three months. The garage door is now half-insulated; I gotta finish that. I have get-togethers away from the house on Sundays and Tuesdays, though that's a little up in the air right now. I've also been keeping up with MLP, and even now, I'm lining up an Applejack puzzle for Tuesday.

Now if you'll bear with me a minute, I'm gonna chase that last bit around. Those of you who've been watching, or otherwise keeping track of, the show, know that the six central characters (known to fans as the "Mane Six") have been, one by one, finding special objects that display rainbow shimmers to the camera. I won't discuss the important details here; I just wanted to note that the first three ponies to find their object were also the first three I made puzzles for, which only had one chance in ten of happening. And I say 1/10 instead of 1/120 for two reasons:

A) because I got the order wrong, so it was six times as likely that I'd match three ponies, and

B) because come on, Twilight's

*obviously*going to be the last one in both lists, so I only really needed to arrange five ponies. (Which, see point A, I didn't do quite right.)

Anyway, I replaced the image for Rainbow Dash's puzzle because some grid lines had been lost to artifacting. And speaking of things I've lost, I had a puzzle written for Fluttershy, but it was... unkind, so to speak. The answer grid was all right, but I wasn't sure how to solve it or how many extra clues to give. Then I put it off so long, I lost the notebook I'd put it in. >(\< So that won't be up for a while, though I'll probably include Rarity's puzzle for completeness, with a color adjustment.

And that's really it for now. See you Tuesday!

Anyway, here's a sudoku variant called X-Sums Sudoku, which I first saw on Para's site (http://puzzleparasite.blogspot.com/).

(Not to be confused with N-Sums sudoku, where N is the

*last*of the N digits to be added. That, incidentally, means not all 36 spots around the grid can have a clue.)

This puzzle type is called "Slalom," and it's probably the best thematic fit so far, a race-themed puzzle for a pony who loves to go fast. Composition was a pain, though. (Had to get it just

*perfect.*)

The object is to draw a loop that goes from square to edge-adjacent square, starting and ending at the checkered square and crossing every hurdle once in order. "Crossing" means starting on one side of a hurdle, taking two consecutive steps perpendicular to it, and ending up on the other side of the hurdle. There are 26 hurdles here, labeled A-Z; ten labels are shown here.

Normally, labels are numbers, and the number of hurdles is given on the equivalent of my checkered square. I felt all right giving the count

*outside*--though actually, there's a little room for that under the grid if this way is too obtuse--because W is so close to the end of the alphabet, so it should be pretty obvious. Also, hurdles are traditionally bracketed on both ends by black blocks (

**which cannot be traveled over,**in the vanishingly slim case that this wasn't guessed), but that obviously crowds grids and would have interfered with the tight theming.

Okay, that leaves AJ, Fluttershy, and Twilight. I can definitely get this done before the end of the TV series, no problem.

Anyway, there are 17 sickles to the galleon and 29 knuts to the sickle. Naturally--well, enough for a recreational mathematician--I wondered if there were amounts of money which, when doubled, transposed the numbers of each coin type. Short answer, yes.

First, a ground rule. Just like you wouldn't say something costs one dollar, two hundred ninety-nine cents, neither should the amounts of wizarding money be named with more than 16 sickles or 28 knuts. Alternatively phrased, an amount of money should always use a galleon when there's an option to use 17 sickles or 493 knuts, and a sickle when there could be 29 knuts.

So now I won't keep you waiting. There are three unit prices (I think), besides free, where the price of one and the price of two use the same numbers of coins in a different order. Find them.