## Solution: 4D. Signs of Change

This puzzle contains a set of fifteen photos of different types of signs, followed by a page with some red dots labelled with seemingly random letters, and a large parenthesised number "(9)" at the top.

It's not at all obvious what to do with the sign photos, so one might first start by attacking the letters on the last page. It's possible to make some progress on this, solving the substitution cipher and decoding the words, but that is not actually work that is useful for solving this puzzle!

The twist to this puzzle which needs to be realised is that all of the Thursday puzzles in this competition are interrelated, and rely on information from other puzzles. In particular, the first step of this puzzle relies on identifying that the fifteen black shapes supplied as part of puzzle 4E. Tangram Mix-up are sized exactly right to fit over one of the sign photos each, blocking out parts of the photo to reveal just a few letters of each sign. In each case, the black shape is oriented the same way as presented in puzzle 4E, without any rotation or reflection. (Solving the substitution cipher and decoding the words on the last page of this puzzle is actually part of solving puzzle 4B. Art Elsewhere, and is described in its solution.)

Overlaid with the correct black shapes, the signs look as follows:

In order, the revealed letters in each sign (with a possibly abbreviated description of the sign) are:

• DCV - RED CROSS BLOOD SERVICE
• WRD - BEWARE GAP WHEN BOARDING
• FIDR - NATHANS FAMOUS SINCE DELICATESSEN CATERING
• FMK - AND LET THY FEET - MIDST KNOWLEDGE
• MQ - MORTAR THE POINT - QUALITY
• MMG - THE MALE DOMAIN HARIDRESSING
• SHNS - FISH N CHIP CRUISE
• SIV - FENWAY PARK - SECTIONS VAN NESS
• PLG - PLUMBING SHEET METAL
• IWD - SWIM BETWEEN FLAGS - DOGS PROHIBITED
• MRK - PUBLIC MARKET CENTER
• TTM - DONT WALK WALK IT'S THAT SIMPLE
• TBD - THE BURGER SHED
• PGSN - STEP INSIDE GENTLEMEN - SEE THIS ATTRACTION
• OCWS - MP POURS COLD WATER ON NUDISTS

Now what to do with these letters? They contain very few vowels, so anagramming seems unlikely. In fact, further progress can only be made by appealing again to the inter-related nature of the Thursday puzzles. Looking at the other puzzles, we can see that the set of fifteen groups of letters, mostly 3 or 4 letters long, matches the set of red highlighted letters in the fifteen unfilled word blanks on the last page of puzzle 4B. Art Elsewhere, and in the exact same order. Entering the letters into the blanks gives us the following:

The next step is to reconstruct the full words! Fortunately, enough letters have been supplied to make this possible, and the results almost unambiguous. A crossword solver such as the one on crosswordsolver.org is helpful here. There are one or two potential ambiguities, but these can be resolved by the fact that the resulting words spell out a sensible, if verbose, message:

DISCOVER WORD FINDER FRAMEWORK COMMUNIQUE IMPLEMENTING ERATOSTHENES SIEVE SPIRALLING INWARD EMBARKING BOTTOM STARBOARD PROGRESSING CLOCKWISE

Interpreted correctly, this means: "Find the word-search grid message by applying the Sieve of Eratosthenes, spiralling inward, starting at the bottom right and advancing clockwise." The Sieve of Eratosthenes is a method of marking composite numbers in a list or grid, leaving behind the prime numbers. The implication here is that we should take the word search grid (from puzzle 4A. All Astir), and label the grid squares with natural numbers beginning with 1 at the bottom right corner, and increasing as we spiral inwards in a clockwise direction.

If we do this, and then cross out the composite numbers (or equivalently highlight the prime numbers), we obtain a grid that looks like the following (in which the letters in prime numbered grid locations are circled):

The highlighted (prime numbered) letters give a message when read in the usual left-to-right reading direction:

FOUR NORTHERN CITIES NORTH TO SOUTH TAKE LETTERS SIX TWO FIVE ONE

Were clearly making progress, but we aren't done yet! This message gives us an extraction method, involving taking specific numbered letters from the names of "four northern cities". But which cities? Now we return to the signs themselves. Most of them are rather anonymous, and it would be difficult to impossible to find out where they are located using the resources available to the average person[1]. However, four of them are relatively famous and in particularly notable locations that can be determined without too much difficulty - simply typing the words into a search engine will suffice. All of the famously located signs are in cities in the northern hemisphere, satisfying the "northern" part of the message:

Ordering these four northern cities by latitude, north to south, as instructed in the message, and taking the indicated letters gives:

• 51° 30' N - LONDON - N
• 47° 36' N - SEATTLE - E
• 42° 21' N - BOSTON - O
• 40° 43' N - NEW YORK CITY - N

The extracted letters spell out NEON. This ties back to the theme of the puzzle, being about signs, as neon signs are a popular and well known type of sign. This helps disambiguate whether this is the answer to this puzzle or one of the others which was raided for information along the solving path. The solution is NEON.

### Puzzle designer notes

The message produced by the Sieve of Eratosthenes had to be designed around the fixed letters in the 4A. All Astir grid dictated by the constraints imposed by puzzle 4E. Tangram Mix-up. (See the design notes for Tangram Mix-up for details about those.) Fortunately, there weren't too many of these letters, and I had degrees of freedom in where to start the Sieve and in which direction to spiral. Starting at the bottom right and spiralling clockwise allowed a minimal interaction of three shared letters between the hidden message of Tangram Mix-up and the one used in this puzzle. In the following figure, the letters fixed by Tangram Mix-up are in shaded blue squares, while the Sieve of Eratosthenes message is in red circles.

Given this, the secret message hidden in this puzzle had to satisfy the following constraints:

• The message had to be exactly 54 letters long.
• The 3rd letter had to be U.
• The 20th letter had to be O.
• The 42nd letter had to be A.

Fortunately, this was not too difficult to satisfy. But of course it also had to be a message relevant to solving the puzzle, preferably involving using more information from the sign photos to arrive at a solution. It was at this point that I tried to manufacture a relevant solution, and hit on the idea of using the four readily identifiable sign locations, and indexing the cities to reveal the solution word. Fortunately, a thematic four-letter word emerged after a little thought, and the secret message could be composed.

The next step was composing the verbose message instructing the solver how to apply the Sieve of Eratosthenes. This message had to be exactly 15 words long, and use mostly words that were relatively long. I wrote a simple message to being with, then used a thesaurus to verbosify it.

Then came the step of extracting the smallest possible set of letters from each of the long words that would make the word identifiable without fatal ambiguities. This produced the set of 2, 3, and 4 letter clusters. And then the next step was to identify those clusters of letters appearing in the signs, in the correct order! I managed to find most of them in my initial set of signs, but ended up having to replace some of them to get some of the combinations. Of course, in replacing the signs, I had to make sure not to remove any of the four northern city signs (fortunately I managed to find letter clusters in each of those), nor introduce any more. Thankfully, I seem to take lots of photos with signs in them, so had enough suitable ones to rummage through.

The final step of making the black silhouette shapes was, relatively speaking, completely trivial. :-)

[1] Actually, it's possible to find the location of several of these signs, as they happen to be photos taken from my own personal collection. A reverse image lookup can locate many of them in my Flickr account, from where it's possible to look at my descriptions and the geotags to determine exactly where I took the photo. However, besides the four northern city signs, all the rest were taken in Australia. I wanted to make sure the puzzle was solvable without needing to locate my Flickr photos, but also if solvers happened to run across them (and potentially get flooded with too much information). Thus the inclusion of "northern cities" in the clue message.