Misc & Plus

DLabelAs you know from my previous How Do You Arrange Your Cheez-Its? posts, Sunshine Cheez-Its are the perfect food. Better yet, some varieties have serving sizes — such as 14, 27, and 29 crackers — that subdivide into great combinations of perfect cubes and perfect squares.

Today’s Cheez-It variety is Scrabble Junior Cheez-It. One serving, can you believe it, is 26 letters crackers! And of course, not only can you arrange a serving into several combinations of perfect squares, you can also arrange a serving alphabetically. You can even do both at once!



A25 1

Figure 1. A serving of Scrabble Junior Cheez-Its arranged alphabetically.
26 = 25 + 1


9 1 16

Figure 2. A serving of Scrabble Junior Cheez-Its arranged as three squares.
26 = 9 + 1 + 16

9 9 4 4

Figure 3. A serving of Scrabble Junior Cheez-Its arranged as four squares of two sizes.
26 = 9 + 4 + 4 + 9

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[Note added 2012/07/26 4:42PM EDT: The Spider-Man designs appear on only one side of each Amazing Spider-Man Cheez-It. Crackers were arranged design-side up.]


My friendly neighborhood Shop-Rite is carrying more and more varieties of Cheez-Its lately, but it didn’t take long to decide on The Amazing Spider-Man Cheez-Its.

The top reasons? 

1. Serving size is a sum of consecutive squares.
2. Flavor is not Monterey Jack.
3. Spider-Man, Spider-Man!




Figure 1. A serving of The Amazing Spider-Man Cheez-Its arranged as a frustum.
29 = 16 + 9 + 4


The Amazing Spider-Man Cheez-It is not the perfect food, but it’s closer to it than Cheez-It BIG Monterey Jack. Spideys taste almost like the original, but they’re a little too hard and crunchy. I can’t imagine eating an entire box in one sitting.


Figure 2. The Pythagorean identity 3² + 4² = 5² was applied to the previous serving.
29 = 25 + 4

The Amazing Spider-Man Cheez-Its are slightly less uniform in size and shape than regular Cheez-Its or Cheez-Its BIG. The crackers chosen for these figures are a more uniform sample than one would expect in a random serving.


Figure 3. Another sum-of-squares serving of The Amazing Spider-Man Cheez-Its.
29 = 25 + 1 + 1 + 1 + 1

Another Steve on the internet has written more about The Amazing Spider-Man Cheez-Its, so I’ll just wrap up with two more arrangements. You should definitely head over to the other Steve’s blog when you’re done here.


Figure 4. One serving of The Amazing Spider-Man Cheez-Its. (Aztec arrangement)
29 = 9 + (1 + 9 + 1) + 9



Figure 5. A variation of Figure 4 with more symmetry.
29 = 9 + (1 + 9 + 1) + 9

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As you know from my last and first How Do You Arrange Your Cheez-Its? post, not only are Sunshine Cheez-Its the perfect food, the serving size of Cheez-Its is 27 crackers, a perfect cube.

As the name suggests, Cheez-It BIG crackers are bigger¹ than Cheez-Its, and a serving contains fewer crackers — 14 instead of 27. While 14 is not a perfect cube, it is the sum of consecutive perfect squares, which is nearly as wonderful.



Figure 1. One serving of Cheez-Its BIG arranged as a pyramid. 
14 = 9 + 4 + 1



Figure 2. One serving of Cheez-Its BIG arranged like a pyramid.
14 = 8 + 3 + 2 + 1



Figure 3. One serving of Cheez-Its BIG arranged unlike a pyramid.
14 = (2 + 3 + 2 + 3 + 2) + 2


Unfortunately, Cheez-Its BIG, the Monterey Jack variety (which I purchased by mistake) is not the perfect food, and I can only recommend it for arranging, not for snacking.



Figure 4. The author (center) ascending a Pyramid not made of Cheez-Its BIG.


¹ “Twice the Size!*”, the front of the box declares.²

² The asterisk leads to a footnote, in smaller type: “*Than Original Cheez-It® Crackers volume.” My rough measurements confirm this. One Cheez-It BIG is about 35% larger than a Cheez-It in each of its larger dimensions, and about 10% thicker.

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Updated June 5, 2012. (Scroll to the end for the update.)

This week’s Numberplay is about a locker room full of, well, lockers – 100 of them, to be exact. The lockers are closed until a janitor (let’s call her Portia) visits the room and opens them all. Then Portia visits the room again and closes every other locker. Later on, she visits the room a third time, opening or closing every third one. And so on, for 100 visits in all.

Head over to the Times, then come back here and enjoy the pictures.

Picture #1. The 100 lockers of the puzzle are represented as a row of tiny squares – red squares for closed doors, green for open. The squares in the top row (all green) show the lockers all open after Portia’s first visit to the locker room.

The second row of squares, alternating green and red, show the lockers after her second visit, and so on.


If you squint at the patterns across the top, I think you can see Portia’s face.

Picture #2. After Portia’s first visit to the locker room, all 100 lockers are open. After her second visit, 50 are open and 50 are closed. Ninety-eight visits later, ten lockers are open. (Just which of the lockers are open in the end is the beautiful result of the puzzle, which you’ll find more easily in the comments to this week’s puzzle than you will by trying to count the tiny squares in the picture here.)

You might wonder – I did anyway – how the number of open lockers changes between Portia’s second and hundredth visit. Here’s a graph.


What a delightfully curious graph! The number of open lockers wanders seemingly aimlessly about the low 50s for a while. It briefly threatens to stay put at 54, then glides down with a few small bumps to a final value of 10.

Intrepid readers can find the Excel spreadsheet I created these pictures from here.

Thanks to Gary Antonick for sharing this great puzzle, which was suggested by Volodymyr Ivanchenko.

Updated June 5, 2012. Back at Numberplay, Gary suggested looking for the number of open lockers sequence at the Online Encyclopedia of Integer Sequences. I didn’t find anything, and I suspect that’s because this puzzle gives different sequences for each initial number of lockers.

So, I put together a SQL Server 2012 script (and posted it at SQL Fiddle here) to generate data for this “number of open lockers” graph for locker rooms with other than 100 lockers.

The graph gets even more interesting when there are more lockers.




One Response to “Numberplay: A Picture is Worth 100 Doors”

  1. Giovanni Ciriani Says:

    Lovely. Seems worthy of publishing in Wolfram’s NKS. Extending the graph ad infinitum, I guess we should see something like an inverted parabola?

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See also:
How Do You Arrange Your Cheez-Its? [#2]
How Do You Arrange Your Cheez-Its? [#3]


Sunshine Cheez-Its are the perfect food, but did you know that the serving size of Cheez-Its is 27 crackers, a perfect cube?

Although individual Cheez-Its are not themselves cubes, or even exactly square, the possibilities are still endless.

Here are two of mine. What are yours?


Figure 1. One serving of Cheez-Its arranged cubically.
27 = 3
× 3 × 3


Figure 2. One serving of Cheez-Its arranged non-cubically.
27 = (9+4) + 1 + (9+4)

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Support Wikipedia

Proud supporter of the Wikimedia Foundation



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Ten years have passed since 9/11. The New York Times put the passage of time into days, and hours, and minutes, and seconds in today’s paper. [A Day That Stands Alone]

Three-thousand six-hundred fifty-two days have now passed. At 8:46 a.m. — the time when the first plane slammed into the north tower of the World Trade Center — 87,648 hours had gone by. Another [*]  5,258,880 minutes. Another [†] 315,532,800 seconds.

For the record, 315,532,802 seconds passed between 8:46 a.m. on September 11, 2001, and 8:46 a.m. today, September 11, 2011. The missing seconds were inserted into our collective timeline by the authority of the International Earth Rotation and Reference Systems Service. One of them passed (largely unnoticed, no doubt) at 6:59:60 p.m. on December 31, 2005 (in New York City), and the other occurred at the end of 2008.

As decades go, this one was as short as they come for us, even with its two leap seconds. Many decades include not two, but three occurrences of February 29th, and all decades beginning between 1972 and 1997 have contained more than two leap seconds in addition to the minimum‡ number of two leap year days per decade.

Nothing is simple.

Steve on the Hoboken waterfront, September 1, 2011.

* Do not be distracted in search of the anaphor. It’s missing, and the issue is not addressed here.

† Another Another is missing its anaphor. Press on, dear reader.

‡ The minimum during our lifetime. The last decade to contain only a single leap year (which was the leap year 1896) ended early in 1904, because 1900 was not a leap year, despite its divisibility by four. The next single-leap-year decade will not begin until the year 2096.

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Or maybe building another wall for stuff I throw to hit. Or maybe throwing a wall at the wall. Hard to say. Tumblr. Yes, that Tumblr.

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You know Facebook is down. You probably didn’t know that Facebook is now a clock.

Service Unavailable – DNS failure

The server is temporarily unable to service your request. Please try again later.

Reference #11.793f748.1285274611.44f235

Notice the number 1285274611 in the error message? That’s the time. Numbers a little over 1.2 billion are almost always times. Unix times. UTC.


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Conflict. Today, my writing was likened to Dan Brown’s, and I’m compelled to demonstrate at least a rudimentary grasp of grammar and its subtleties.

I write like
Dan Brown

I Write Like by Mémoires, Mac journal software. Analyze your writing!

Interlude. Let me explain how I arrived at this conflict; skip to the dénouement if the travelogue begins to bore you. [Note to self: look up or else coin the adjectival form of interlude; consider interludinous, interludinal, interludinary, interludine.]

The comparison of my writing with Dan Brown’s occurred earlier today, while I was visiting I Write Like, a momentarily amusing web¹ site at http://iwl.me. I arrived there from this CONJUGATE VISITS post (sorry, but its author yells the title). I happened onto CONJUGATE VISITS while looking up “supposably,” which I learned today is a word (note the absence of scare quotes around “word”), as opposed to a “word,” which would have been my first guess.

The next step back is a tad embarrassing. I only realized where I’d been before looking up supposably when I retraced my steps for this blog post; I’d gotten the idea to look up supposably from this article on the web site of Reader’s Digest, a generally icky place I wouldn’t have visited intentionally. A tweet from Phil Jimenez led me to the Reader’s Digest article (more specifically a bit.ly URL in the tweet, and I submit disguise-by-shortening as my excuse).

I don’t recall whether I read Phil’s particular tweet before or after I noted that he and I shared exactly one Facebook like, Dan Savage. That was no surprise, given what (or who? It’s a fictional character, so I’m not sure.) led me to Phil’s Twitter stream in the first place — Kevin Keller. Kevin, as you may know, made his appearance in Veronica #202 today; while I’ve yet to get my hands on the issue, I’d caught wind of it from Google News and consequently searched Twitter for the latest buzz, finding Phil, then Reader’s Digest, then supposably, then CONJUGATE VISITS, then I Write Like. In summary,

  • I Write Like, from
  • supposably, from
  • Reader’s Digest, from
  • @philjimeneznyc, from
  • Kevin Keller, from
  • Google News, from
  • daily routine.

Dénouement. On to my demonstration. Consider the following sentence, which I found on Amazon in a one-star review of CONJUGATE VISITS’s authoress June Casagrande’s book, It Was the Best of Sentences, It Was the Worst of Sentences, here.

Copernicus was thrilled when he discovered that the earth revolves around the sun.

Casagrande and the reviewer both prefer this to “Copernicus was thrilled when he discovered that the earth revolved around the sun.” I on the other hand, presently compelled to say something about grammar, offer an even better sentence.

Copernicus was thrilled to discover that the earth revolves around the sun.

The proposition of Casagrande’s sentence (either version) has two parts. Deconstructing the sentence rigorously, it states first that Copernicus was thrilled, and second that Copernicus’s² thrill occurred when he made his now famous discovery. However, the second part of the proposition is perplexing, if only slightly. If the writer had stopped after “Copernicus was thrilled,” I’d have felt cheated, but because she’d failed to explain why he was thrilled, not because she’d failed to explain when he was thrilled. Emotions interest readers because of their why, not their when.

For most readers, I’m sure the second part of the sentence as written sufficiently explains the why. Similarly, if the “thrilled when” sentence were part of an SAT reading comprehension question, the “correct” answer to Why was Copernicus thrilled? would be a) Because he discovered that the earth revolves around the sun., not d) It’s impossible to determine from the reading. But why explain “why?” indirectly by explaining when? The turn of phrase “thrilled to discover” isn’t the only choice — one might say “thrilled by his discovery” or “thrilled to have discovered,” but it’s the best choice, and this is my blog. Also, I might have answered d) to the SAT question, especially if I knew I’d get to argue with a teacher about it later. I don’t brag about my SAT English score, and for good reason.

Epilog. Dare I paste this blog post into I Write Like? And if I do, then post the result here, then paste it in again, will the result be the same, and if not, and I repeat the process… [Update: The result is … H. P. Lovecraft. I’ll leave it at that. Tear from the fabric the threads that are old!]

I write like
H. P. Lovecraft

I Write Like by Mémoires, Mac journal software. Analyze your writing!

Postscript. You, dear reader, are a mensch for getting to this point. Let me know how I can return the favor. You are almost as much of a mensch as Itzik, who hired me as an editor … twice, the second time after knowing how I go on about things like this.

¹ By writing web and not Web, I comport with one of the “Significant Rule Changes” in the latest edition of The Chicago Manual of Style. The interested reader (which is to say You, because you’ve read this far into my footnote) can find the full list here. This footnote is not an endorsement of The Chicago Manual of Style.

² Ibid. Among the Significant Rule Changes are rules on the possessive forms of two kinds of names: those ending with an unpronounced “s” and those ending with an “eez” sound (in the latter case presumably when the name also ends in “s,” because there can’t be any debate on possessives like Lise’s). Copernicus falls into neither category, and I don’t know the latest rule on his possessive. My rule is to always add ’s to form a possessive (as in This is Steve Kass’s blog.) except maybe for Jesus, Moses, and princess. Even for them I’m not certain what I’d do, but they don’t come up in my writing much.

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