Back in March, I saw a video featuring two puzzles about prisoners wearing hats. I subscribed to Matt Parker's channel either shortly before or shortly after; I don't remember which. Anyway, here's a series of LiveJournal cuts.

**( Here"s the video!Collapse )**

Because he already gave a non-bumbling explanation, so I shouldn't have to go through all that work. Also, I will be referring to this video, so even if I'd summarized the puzzle, this is required viewing.

**( Here"s my immediate thought, and a grinding derivation of the solution for 3 prisoners in 4 hats!Collapse )**

But I admit I didn't actually write down this table, or construct it in my head. Not sure if that makes my hitting upon the general solution more or less impressive, but there you go.

Without writing anything down, I felt stuck. How could I code twelve possible ordered pairs (much less 60 triples, 360 quadruples, or more longer permutations) into one binary guess? And I knew it had to be binary, because not only would guessing anything other than a possible hat 100% doom the Back prisoner, the rule of not repeating a guess would doom whoever was really wearing that hat.

Of course, the puzzle never said the guess had to be in the range of possible hats. One commenter suggested that the Back prisoner add up all 999 numbers of the other prisoners, and this would absolutely save 999.0 prisoners. But I remained certain that with the right approach, I could save 999.5. If no one gave an answer they knew was impossible, then everyone would have two possible hats when it was their turn, and one hat would always be their real hat. Just, how could I choose permutations that were sensitive to every slight rearrangement, communicated that sensitivity only by alternating the Back prisoner between two equally risky guesses, and could be extended to any arbitrary group of prisoners?

**( And the answer!Collapse )**

Then again, if you're not all perfect logicians, or if the mastermind changes his mind, the lot of you are in trouble.

- Current Mood: exhausted
- Current Music:The Stand-up Maths theme

Anyway, it's still Tuesday for another 48 minutes here, so I hope you're ready for a puzzle!

I saw an interesting license plate the other day, and I'd like to tell you about it. It consisted of three letters followed by three digits, which isn't interesting by itself, but bear with me.

The letters were such that each of them, in order left to right, appeared in the spelled-out name of the corresponding digit. What's more, with the standard A=1, B=2 substitution, the product of the three letters' values equaled the number formed by the three digits!

Now to get vaguely specific. I won't tell you the three-digit number, and neither will I tell you if knowing the number would be enough information to determine the three letters and their order. But if I answered the second question, you could determine one of the plate's digits (and maybe its position, but maybe not). What's more, if I told you that digit's corresponding letter, you could tell me the entire plate!

So what is it?

I've been a little busy. Not busy with exactly the things I'd prefer, I admit, but some of them are what I want. Like, I'm doing a wimpy (by volume) version of NaNoWriMo; I'm hoping to finish a few chapters by the end of November.

On to the puzzle! Maybe you have heard of Equestria Daily. Last month, they finally got mascots, Spotlight Splash and Rocket Tier. When I read that Spotlight likes solving puzzles, that lent me the impetus to write this puzzle, a pony-themed alphacipher.

In a normal alphacipher, each letter has been assigned a distinct integer value from 1-26, and several words are provided along with their values as determined by adding together the values of their letters. For example, with the common A=1, B=2... Z=26 assignment, EXAMPLE has a value of 5+24+1+13+16+12+5=76. (Yes, both E's count.)

However, I performed some different calculations. The rule I used does coincide with the normal addition in many predictable cases, but... well, my Puzzle Hunt experience is keeping me from outright telling you the rule. If you find that "figure out the secret rule" approach to be too difficult--or obnoxious--here's the rule in rot13:

Rnpu anzr vf gb or fcyvg vagb cnegf bs rdhny yratgu; gur yratgu va rnpu pnfr vf gur ynetrfg cevzr ahzore gung jvyy jbex. V gbbx gur fhzf bs rnpu cneg naq zhygvcyvrq gurz gbtrgure sbe gur tvira inyhr. Sbe vafgnapr, gur rknzcyr jbhyq jbex gur fnzr, naq gur qvfp wbpxrl hfrf gur cebqhpg bs sbhe fhzf bs guerr yrggref rnpu.

Here's the unadorned version:

APPLEJACK..........27550 BON BON.............2401 FLUTTERSHY..........4148 LYRA.................510 MUFFINS...............68 OCTAVIA..............108 PINKIE PIE.........65379 RAINBOW DASH.........186 RARITY..............2736 SPIKE.................49 TIME TURNER.........5082 TRIXIE..............2624 TWILIGHT SPARKLE..369279 VINYL SCRATCH....3270960 ZECORA...............663

How much is EQUESTRIA?

In semi-related news, as mentioned above, I am attempting to do a low-intensity version of NaNoWriMo, with a crossover called "Dan Vs. King Sombra" which, irrespective of obviousness, is a crossover between "My Little Pony" and the no-longer-in-production cartoon "Dan Vs." My original goal was to write something for the novel every day. I technically failed yesterday, but I fully intend to keep writing.

- Current Music:Weird Al, "Hardware Store"

Hi there. With the dawning of a new year, I've decided to make some changes. Starting tomorrow, I'm going to try to post on LiveJournal every day.

I'm certainly keeping up my puzzlemaking, and I'd like to point out that some of my creations are going up on Grandmaster Puzzles. Moreover, I intend to put out some life updates here, along with puzzles that don't quite fit over there.

Happy New Year, everyone!

- Current Music:"Party in the USA"

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!