Friday Puzzle #114 - The Count's Count
Type: Counting Puzzles; Spot the Differences
USPC History: Counting puzzles of various sorts have appeared in 2000, 2004 (x2), 2005, 2007, 2009, 2010. Usual constructors include Scott Kim and Nick Baxter.
Spot the differences (sometimes spot the matching pairs) have appeared in 2000, 2001, 2002 (x2), 2003 (x2), 2004 (x2), 2005 (x2), 2006 (x2), 2007 (x2), 2008, 2009, 2010. The usual constructor is Patrick Merrell although Nick Baxter has also made some, including some "collaborative" puzzles where Nick has transplanted Patrick's mice to weird new places.
Strategy/Notation tips: Not much strategy advice for Spot The Differences puzzles. Notationally, just as with a word search, I prefer to use a colored pencil to mark differences, certainly on the most challenging Spot The Differences puzzles like the 90-95 difference ones in Nikoli's magazines. Otherwise, an aggressive circle or arrow in a dark color could sit on top of that last difference after you've looked at the puzzle for a half hour. On the USPC, the spot the differences puzzles have grids that define the answer entry so when I find a difference I fully shade the square to black it out from my view in the future, and to make the answer extraction trivial as well. If a whole side of the puzzle is not shaded, there are likely differences to be found there as most puzzles spread differences throughout the drawn space. I almost always start the USPC by solving this puzzle as the test prints. I'll have ~6-8 of the differences in mind before I go to paper, and get the rest soon after.
For counting puzzles, there is no fool-proof strategy as there is never a certainty when you have the answer. But there are some tricks. The first is to be as systematic as possible when counting things. If you are counting squares with unit distances, separate the 1x1 and 2x2 and 3x3 and so on counts from each other and double check each independent tally. I like to mark a specific corner (say the upper left corner) of squares when counting like this to then visually scan the puzzle again quickly at each size. An independent count focused on lower right corners would be the check tally. A similar approach can be used when counting square-like objects such as the L in Count Me In in 2005. There I used numbered arrows to indicate the size and missing corner of all the placements.
If the count is something else like triangles, you may want to focus on counting all triangles from the "top" point, if such a thing exists, or count all triangles from all vertices to get 3N of the answer. If you don't have a multiple of 3, you have made a mistake. Even if you have a multiple of 3, you might have made a mistake. There are often "tricks" with USPC counting puzzles. In 2010, this trick was recognizing only concave pentagons existed in the puzzle and marking the count specifically at the potential concave vertices.
Counting puzzles are high risk puzzles and for most solvers are not worth the likelihood of a -5 point tally. So unless you have to solve the counting puzzle as there are no other puzzles you feel you can solve, the best strategy is probably to skip it entirely.
Comments: I've discussed in other entries how observational puzzles are good, approachable puzzles for solvers of all ages and thus they deserve some small fraction of the USPC space. However, if you've read this blog long enough, you'll see I greatly prefer puzzles like spot the differences where you will know when you've spotted everything from a check sum in the puzzle versus counting puzzles where you are most often 1 or 2 off of the correct answer without knowing it. Counting puzzles will always be my nemesis on this test.
One reason I dislike counting puzzles is because they are almost always worth -5 points for me if I have the time to attempt them. Every year I've had a debate with Nick Baxter on the scoring of the USPC and how penalties for wrong entries in some puzzles don't make sense. The penalty is meant to discourage random guessing but for most answer keys I'd say this isn't a potential problem. Getting a reduced score (and not just a zero score) for a mistake seems too punitive. Sudoku puzzles do not need penalties. Battleships puzzles do not need penalties. But then there are puzzles where "free guesses" would have a reasonably high expected value (of a half point or higher). Fill-ins where you are entering a missing word from a small list probably fit, as do the spot the difference variants that involve choosing one from a set of 3-10 possibilities. I would state a specific penalty for these puzzles that would scale to be greater than - probably double - the expected value of a guess.
Counting Puzzles sit very much in the middle of these extremes. An honest attempt will often get close to but not be at the right number. Such a total might even win a "how many jelly beans are in this jar" contest at a state fair without being correct. Counting puzzles are hard to get right. Constructors make mistakes on these all the time. I wonder if Nick Baxter chooses some of the ones he does (mostly from Scott Kim's puzzle calendar work) because of mistakes he's found in the expected solution.
So the most likely result of an honest attempt will be a close answer, but on the USPC a close answer means negative points. I've often wondered whether a variable penalty based on the submitted value is the right fit here. The distribution from 2007 is maybe best to build from for this argument. There, a 37 deserves full marks, but an answer between 35 to 39 probably probably deserves no penalty, and an answer grossly off (like "4") probably deserves -5 as there is no obvious investment of time to reach a reasonable number from such a wild guess. A solver who spends 20 minutes to get to 36 (like me) has already been penalized enough by spending time to not earn points and does not need an additional penalty set up to discourage guessing when 36 on that puzzle is not likely to be a guess.
There are many types of counting puzzles; some seem very arbitrary and taxing like the three overlaid circle stencils in Circular Logic in 2004. Others have a workable approach rooted in mathematics like the Window Pain puzzle in 2009. I certainly prefer the latter, but hold my breath every year until the answer key is released and I know what my result is on that puzzle. Counting puzzles have even been the source of inside jokes between the test organizers and myself over the years. Notice the concave versus convex polygon comment in the answer entry there. That question comes up all the time at WPC instruction meetings.
While I practice but do not actively seek out counting puzzles, I do play Spot the Differences puzzles much more frequently. A lot of online flash games use a difference finding gimmick which is one quick way to get used to tracking visual information with both eyes at different spots at once. And as mentioned above, I do sit and go through the STD marathons in Nikoli magazines with 90 or more differences to find. They have some really nasty differences in there from time to time - quite subtle until you actually see them - and perhaps the practice helps me expect where that line segment will be just a smidgen longer.
About this puzzle: Since these observational puzzle types occur so frequently, I had to give a shot at constructing them. But to assist the Counting Puzzle, I felt I would use the Spot the Differences as a check-sum on the other. You can certainly solve the Counting puzzle as is, and see if you get the correct number of squares. But if you ID the two "identical" Count images below, the shared number in them will also tell you the answer. Either route - counting squares or spotting differences - will earn you credit here.
Count the number of squares that occur in the image below. The shading in the image is for aesthetic purposes only, representing a set of rectangular cards spread over a square tiled surface.
Not including the variable "counts" in the speech bubbles below, and ignoring the mirroring of the images, there are two identical images below and eight others with one small change from the rest. Identify the identical Counts. Enter the shared number in their speech bubbles as your answer (this value is also my expected answer for Counting Cards).