Friday, December 14, 2012

Curious Cuboids

A few more diversions from Insomnia's frozen shore

Recently I was thinking about rectangular parallelepipeds, also known as cuboids. A cuboid is a simple box shape. Think of a shoebox, a cube, or the black monolith in the film 2001: A Space Odyssey.

A cuboid has 12 edges with lengths x, x, x, x, y, y, y, y, z, z, z, z. By analogy to triangles, I call a cuboid scalene if x, y, and z are all distinct, isosceles otherwise. I further classify isosceles cuboids as being rod-shaped (x = y ≤ z), tile-shaped (x = ≥ z), or both (x = y = z; a cube).

Isosceles cuboid, rod-shaped
Isosceles cuboid, tile-shaped

To find the volume, multiply length times height times width. To find the surface area, compute the area of each of the six faces, then add them all up. For a linear measure, let's define the edge sum: length plus width plus height.

Below are some hopefully amusing questions about all of this. Also, I made a grainy animation that relates to all this. (It will make more sense after having thought about the questions below.)

Some of these questions can be settled with visualization, and some by simple arithmetic; others are less simple. One day I'll post some further observations on these the meantime, enjoy! And feel free to put any answers or thoughts in the comments.

Volume and Surface Area. The (6, 6, 6) cuboid is notable in that its volume is 6 × 6 × 6 = 216, and its surface area is (6 × 6) × 6 = 216.

1) Can you find another example of a cuboid with volume and surface area numerically equal when expressed in the same system of units? Is your example scalene, rod-shaped, or tile-shaped? Can you find an example of each?

Surface Area and Edge Sum. The (½, ½, ½) cuboid is notable in that its surface area is (½ × ½) × 6 = 3/2, and its edge sum is ½ + ½ + ½ =  3/2. 

2) Can you find another example of a cuboid with surface area and edge sum numerically equal when expressed in the same system of units? Is your example scalene, rod-shaped, or tile-shaped? Can you find an example of each?

Edge Sum and Volume. The (Sqrt[3], Sqrt[3], Sqrt[3]) cuboid is notable in that its edge sum is 3Sqrt[3], and its volume is 3Sqrt[3].

3) Can you find another example of a cuboid with edge sum and volume numerically equal when expressed in the same system of units? Is your example scalene, rod-shaped, or tile-shaped? Can you find an example of each?

In each case above, two measures were equal.

4) Are there any cuboids for which all three measures are numerically equal when expressed in the same system of units?

A few more existential questions...

5) Is there a cuboid with a volume of 1 cubic inch and a surface area of a million square inches?

6) How about a cuboid with a volume of 1 cubic inch and a surface area of a millionth of a square inch?

7) Is there a cuboid with a volume of a million cubic inches and an edge sum of 1 inch?

8) Is there a cuboid with volume 80 cubic inches, surface area 122 square inches, and edge sum 16 inches? 

Finally, some optimization problems. In each case, the desired maximizing/minimizing cuboids exist and have whole-number edge lengths.

9) Of all cuboids with surface area 48 square inches and edge sum 9 inches, which has greatest volume? Which has least volume?

10) Of all cuboids with volume 72 cubic inches and edge sum 14 inches, which has greatest surface area? Which has least surface area?

11) Of all cuboids with volume 162 cubic inches and surface area 234 inches, which has greatest edge sum? Which has least edge sum?

Sunday, August 5, 2012

Cents on the Dollar, Some Attempts

How much of a dollar bill can you cover up using change for a dollar? The coins must not overlap one another.

(See previous post and the one before that.)

I made a few attempts at this, see pictures below.

Version 1, wide-open. Here I covered the dollar bill with 45 pennies, going for a hexagonal lattice. I'm guessing this covered a percentage of the dollar in the mid- to high-80's. (One easily shows that an infinite hexagonal lattice covers a fraction Pi*Sqrt[3]/6, or a little less than 91%, so I'm assuming the below is pushing up against that limit.)

Version 2, intermediate. I nudged the configuration above in order to make room for the additional coins to touch the dollar bill - that seemed to force a decision to remove five of the pennies and replace the nickel with a dime.

I did some quickie image analysis on the above awful picture to give me some inputs for a Monte Carlo calculation of the covered area:

The result was just under 85% coverage. I didn't do a sensitivity analysis. I'm guessing this configuration is low- to mid-80's.

Version 3, tight. Here, my attempt uses 8 dimes, 15 pennies and a nickel. According to the spreadsheet, this configuration covers just over 64% of the dollar.

It would be nice to be able to replace that nickel with 5 pennies - according to the spreadsheet, that would bump the area up to 74.6%. But I wasn't seeing quite enough room to do that, and truth be told I have put a lot more time into blogging this puzzle than I've spent trying to solve it. Feel free to blow these attempts out of the water!


I thought of this puzzle after daydreaming the following exchange between a bank teller and an eccentric customer:

"May I help you?"
"I'd like change for a dollar."
"Here you go!" (Hands the customer four quarters.)
"Why, you cheat! I had 16 square inches' worth of currency and you gave me back only 3 square inches' worth!"
"I'm sorry, sir. Here are 7 dimes and 30 pennies instead."
"OK, close enough...."

I thought it was amusing, this image of a person valuing an exchange along a totally irrelevant dimension. As if one were to share a long kiss with somebody, and then become upset because the other person took a different number of breaths than you did.


The above puzzle takes the form, "How many X can you cover with Y?" I'm sure there a lot of problems and solutions of this nature to be found in the literature on circle-packing. I don't know that literature, but a quick search just now turned up some neat pictures here.

Thursday, August 2, 2012

Cents on the Dollar, Ctd

How much of a dollar bill can you cover up using change for a dollar? The coins must not overlap one another.

Yesterday I played the game three times, under three different conditions:
Version 1: Wide-open. In this version, you don't have to use all the coins. Equivalently, we might characterize things by saying that while you do have to place all of the coins adding to $1.00 on the table, not all of the coins on the table have to touch the dollar bill. Also, those coins that do touch the bill are allowed to protrude beyond its edges.
Version 2: Intermediate. All of the coins adding to $1.00 have to touch the bill. Coins are allowed to protrude beyond the edges.
Version 3: Tight. All of the coins adding to $1.00 have to lie within the boundary of the bill. 
(The spreadsheet is appropriate for scoring attempts that meet the constraint in Version 3.)

On Monday I'll post a picture of my attempt at each version.

Tuesday, July 31, 2012

Cents on the Dollar

How much of a dollar bill can you cover up using change for a dollar? The coins must not overlap one another.

A not-very-good effort is pictured below. The four quarters only cover about 18% of the dollar bill's area.

Send me a picture of your best effort!

N.B., I made a spreadsheet that you can use to score your attempts. Edit the blue values in the spreadsheet to reflect how many coins of each type you used. Please don't edit the formulas, or others won't be able to use the spreadsheet. (Let me know if there are errors.)


Tuesday, June 5, 2012

Incurable Romantic

With my wedding anniversary coming up, I consulted some of those lists of traditional gifts. We're not at the gold or silver level yet, so I see there's the bronze anniversary, the paper anniversary, the leather anniversary (sounds spicy!), even the salt anniversary. (Sort of spicy, too, I suppose.) Today I was doodling on the computer, so apparently this year is the computer graphics anniversary.

Higher-resolution file here (.wmv file, 13.5MB).

To make this shape, I took forty-one circles of latitude on a sphere and replaced each one with a heart. The scale factors of the hearts are the same as the scale factors of the corresponding circles of latitude. In that sense, the shape is an example of a generalization of the sphere, in which a stack of similar circles is replaced by a stack of other similar shapes. (My wife and I were talking about this subject the other day.)

The parametric equation of a heart is from this page. Some other nice heart ideas are here.

(Don't worry, honey, I also bought you a non-mathematical present!)

Saturday, April 21, 2012

Triple Double Revisited

I haven't had many solutions to the Triple Double, so I'll put a more fun(?) version of the puzzle here.

In this form, it's the kind of thing where you can do some of them during a quick coffee break.

In each of the following cases, rearrange the letters to form a word.

(Note, a number of these words are hyphenated.)

























Saturday, April 14, 2012


Recently I happened to be looking at the algebraic expression xy + yz + zx, when suddenly the variables in the expression morphed before my eyes into the word XYYZZX. Which is not a word, of course (but it sounds like one that rhymes with "physics").

This led me to wonder: Which words are "triple-doubles"? By a triple-double I mean a six-letter word with exactly three distinct letters that are each used exactly twice. An example of a triple-double is SESTET.

The word SEEDED, on the other hand, is not a triple-double. While it does have six letters, and while it does have exactly three distinct letters, the letter E is used three times (instead of twice) and the letter S is used once (instead of twice).

1) Can you think of any triple-doubles?

2) Can you think of any quadruple-doubles? (A quadruple-double is an eight-letter word with exactly four distinct letters, each letter used exactly twice.)

3) How about a quintuple-double? (Ten-letter word, exactly five distinct letters, each letter used exactly twice.)

This morning I was able to find a number of triple-doubles and a few quadruple-doubles before curiosity got the best of me and I went to the computer. Just FYI, the computer did actually find a few reasonable quintuple-doubles, including at least one really nice example.

Anyhow, if people send me any of the above, then I'll summarize in a future post. Feel free to put examples in the comments, or send me an email!

P.S. In case you weren't sure - and I wasn't, even though I had thought of the word itself and was pretty sure it was a word - a sestet is a group of six lines at the end of a classical Italian sonnet. And by the way, if you click that link and read the poem they have there, you'll see a nice triple-double in the sestet.  ;-)

P.P.S. Everything a hoops fan might want to know about double-doubles, triple-doubles, quadruple-doubles, quintuple-doubles, and related statistical feats is here. As you might guess, Hakeem Olajuwon pretty much cleans up in these categories...although David Robertson also managed to demolish my Pistons pretty spectacularly back in 1994.

Monday, March 12, 2012

Snowdrops, Christmas Lights, Rush Limbaugh, and the Director Circle of an Ellipse

Our snowdrops bloomed on Sunday...the sun has been warming the earth. It was as good a day as any to take down the Christmas lights.

There are places where leaving your lights up until March would get you in trouble with the neighbors. In Vermont, we live and let live. I chaired my town's Planning Commission for a year; this is the group responsible for the town's zoning bylaws. One time, when we were debating the issue of junk cars in our town, a representative from the state government showed us a photo of a large junk car collection in the northern part of the state. This photo had been taken from space.

Anyway, speaking of space, I tend to take an astronomical view of things; I figure I'm not really late if the decorations come down before the vernal equinox.


I heard that Rush Limbaugh's sponsors are leaving in droves. People like Michael Kinsley, Andrew Sullivan, and Bill Maher have stressed that living in a country with free speech means that you're going to be made uncomfortable sometimes by what people say, and that you should fight speech with speech, not boycotts or campaigns.

The story here seems to confirm their fears.

I agree with Kinsley et al. about campaigns that would topple someone for stating their views. And I agree that if you don't like Rush, then you shouldn't listen to him, and you should counter what he says with better arguments and better rhetoric. If enough people stop listening to him, his advertisers will leave anyway.

But on the other hand, I also don't see how Rush Limbaugh's right to free speech obligates me to subsidize him. If we all have a duty to support Rush Limbaugh, then maybe Congress should pass a federal sales tax to keep his radio show running.

And what about people who aren't part of any "campaign"? If I decide on my own to send a tweet to ProFlowers saying I'm going to shop with their competitors as long as they advertise on Rush's show--not that I did, mind; I don't have a Twitter account--then I haven't been part of any campaign or attempt to "manipulate the system." So far, Kinsley, Sullivan, and Maher have argued against organized activity. What is the argument against individual consumer activism?

I suppose the argument is that consumer activism is good when it counters harmful actions--as when someone refuses to buy products from factory farming, out of a judgment that factory farming is a set of harmful actions. But consumer activism about speech acts gets into troubling territory.

Or maybe the argument is that the logical endpoint of individual consumer activism against media entertainers is a less vibrant media-tainment sector. If a small percentage of consumers, even without acting in concert, can spook advertisers through the power of social media, then we might not get to see people like Rush Limbaugh and Bill Maher.

That's the argument against tyranny of the minority. But despite what it says in the MediaMatters story linked to above, I don't think that's what's happening to Rush. I think what's happening to Rush is that a large percentage of consumers has finally decided  that paying his salary makes them feel dirty. It seems to me that an entertainer who enters into this particular territory has made his own bed.

Rush Limbaugh has a large audience--large enough, I suppose, to keep him in business even without sponsors. I imagine he could take his show to a subscription service on satellite radio. Or he could run for the Senate, or write a weekly newspaper column, or become a Republican speechwriter, or publish books. I'm not really worried about Rush Limbaugh's free speech rights. But I do wonder about the longevity of a business model founded on being an asshole.

Having said that, I'm impressed by the distinction I made earlier (if I do say so myself) between individual consumer activism about actions vs. individual consumer activism about speech acts. Perhaps with regard to the latter, we need moderation. So maybe the idea is that even if you would rather not listen to the KKK radio hour yourself, then even so, you shouldn't be worrying about whether companies you do business with are advertising there. Call it the "ignorance is bliss" theory. If it's true, then it would be yet another case of one of the paradoxes of democracy, that moderation is needed in order to enable the immoderate to flourish.


I was doodling the other day and "discovered" a nice property of ellipses. The puzzle I'd set for myself during some downtime (I think it was when my wife was outside taking down the Christmas lights) was to determine the locus of points from which an ellipse subtends 90 degrees. I did some algebra (here), and lo and behold, the locus is a circle! Here is an animation:

This property of ellipses gets forgotten and rediscovered periodically, at least to judge from this exchange on MathForum involving John Conway. Wikipedia knows about it. Here in case you're interested is a sketch of a proof from synthetic geometry, from a site with a wealth of geometry problems.

Given the equation of an ellipse and the equation of a line, my derivation simply demands that these simultaneous equations have a double root (for tangency), which determines two possible slopes, corresponding to the two tangents. Requiring these tangents be orthogonal leads to the equation of the circle. You can try this method with the parabola y = ax^2 to see how it works. (The algebra is less involved in this case.) If you do the parabola problem, you'll see why the circle in the above animation is called the director circle of the ellipse.

Friday, March 2, 2012

Some Solutions to the Quadruple Vowel Puzzle

There's nothing like a very long plane ride for getting work done. Here then are some answers so far to the quadruple vowel puzzle, including answers from the comments and answers from emails people have sent me. (N.B., occasionally I chose not to include a human solution if it "just didn't look as nice" as other solutions of equal or lesser length, or if it would have made the list seem too long. So the list is an editorial project to that extent.)

Also included below, in a few cases, are some notable answers from an online dictionary file available here (.txt file). An asterisk denotes what one might consider a "computer victory."

Word length isn't the only criterion one might use to judge the quality of a solution. For example, word frequency might be used to break ties. (On that view, SOMEONE is better than HONOREE.) Aesthetics might also enter into it; DOVECOTE is just a more beautiful solution than LOOSENED. One solver proposed a quality scale that involves linguistic concepts such as the number of morphemes in a word. Perhaps we can ask him to identify the "nicest" solutions according to that metric in the comments section.  Finally, one might argue that in the case of this puzzle, very long words are as good as very short ones, seeing as how piling letters atop one another gets harder and harder to do without adding a disqualifying vowel. (Finding such words is really CATCH-AS-CATCH-CAN...just ask any FENCINGMISTRESS or CANVAS-STRETCHER.)

In any case, the quality of the solutions was incredibly high.  Many or most of the words listed below were found by two or more solvers. Moreover, you'll see that there are few computer victories. Short words conforming to the given patterns are evidently rare, yet a small group of us found many of the best possibilities. Perhaps whenever we read, we are consciously or subconsciously attending to these rather "odd-looking" words and storing them away in our minds.

Thanks for playing!


Solutions are presented in the following format: Humans first, then computer in a few cases. Each list is sorted by length, then sorted alphabetically within length (modulo clerical errors).

*1. BALACLAVA (9), CATAMARAN (9), JACARANDA (9). Notable computer answer:  MAHARAJA (8)


3. AIRMAIL (7), MAINTAIN (8), MAINSAIL (8), BRAINIAC (8, slangy perhaps).

*4. BOATLOAD (8), MACAROON (8). Notable computer answers: AVOCADO (7), BAZOOKA (7)


6. ANYWAY (6), PAYDAY (6)



*9. HONOREE (7), DOVECOTE (8), LOOSENED (8), METRONOME (9), MONOTREME (9, jargony perhaps; anagram alert!). Notable computer answers: BOOTEE (6, if you can stomach that spelling), SOMEONE (7)

10. QUEUE (5), BURLESQUE (9), UNDERUSED (9), UNRETURNED (10). Notable computer answer: UKULELE (7)

11. BYE-BYE (6, if you can handle the hyphen), DYSENTERY (9). Notable computer answer: WHEYEY (6, archaic perhaps)


13. COITION (7, jargony perhaps), OMISSION (8), POISONING (9), PROBIOTIC (9, used to be jargony but not really anymore), SOPORIFIC (9). Notable computer answer: OPINION (7)

*14. UNSUITING (9), FUTURISTIC (10), SUBSTITUTING (12), UNFULFILLING (12). Notable computer answers: JIUJITSU (8), RUBIDIUM (8, jargony perhaps), UMBILICUS (9, jargony perhaps)

15.YIPPILY (7, judgment call; "Well, it is how small dogs bark," argues the solver), WILLY-NILLY (10), NITTY-GRITTY (11). Notable computer answer: TYPIFYING (9)


17: VOUDOU (6, unusual spelling), OUTPOUR (7), COUSCOUS (8), HUMOROUS (8), OUTBOUND (9), GLOBULOUS (9)

18. YO-YO (4), BOY-TOY (6, maybe a little neologistic), ROLY-POLY (8)

19. MUUMUU (6)


21. For YYYY, one solver suggests 

SYZYGYLY (suh-zi-ji-lee): In such a manner as to cause three celestial bodies to align. "The supersitious began to panic as the planets moved ever more syzygyly across the heavens."
(Seeing as computers lack humor, we win this one.)

Previous word puzzles here, here, here, and here.

Sunday, February 19, 2012

Quadruple Vowel Puzzle

UPDATE: In the original post, I forgot to clarify the rule about Y's. See below.

Here's a puzzle I made up recently to pass the time on an airplane. Solutions welcome in the comments - I'll post my solutions later on.

(Read through to the bottom for notes and suggestions.)

1. AA/AA: Find a word with four A's, and no other vowels.

2. AA/EE: Find a word with two A's and two E's, and no other vowels.

3. AA/II: Find a word with two A's and two I's, and no other vowels.

4. AA/OO: Find a word with two A's and two O's, and no other vowels.

5. AA/UU: Find a word with two A's and two U's, and no other vowels.

6. AA/YY: Find a word with two A's and two Y's, and no other vowels.

7. EE/EE: Find a word with four E's, and no other vowels.

8. EE/II: Find a word with two E's and two I's, and no other vowels.

9. EE/OO: Find a word with two E's and two O's, and no other vowels.

10. EE/UU: Find a word with two E's and two U's, and no other vowels.

11. EE/YY: Find a word with two E's and two Y's, and no other vowels.

12. II/II: Find a word with four I's, and no other vowels.

13: II/OO: Find a word with two I's and two O's, and no other vowels.

14: II/UU: Find a word with two I's and two U's, and no other vowels.

15: II/YY: Find  word with two I's and two Y's, and no other vowels.

16: OO/OO: Find a word with four O's, and no other vowels.

17. OO/UU: Find a word with two O's and two U's, and no other vowels.

18. OO/YY: Find a word with two O's and two Y's, and no other vowels.

19. UU/UU: Find a word with four U's, and no other vowels.

20. UU/YY: Find a word with two U's and two Y's, and no other vowels.

21. YY/YY: Find a word with four Y's, and no other vowels.


I was not able to solve #20 or #21.

"Two O's" means exactly two O's, not "two or more." It does not mean that the two O's have to be consecutive (although consecutive is allowed of course).

UPDATE: I ought to have noted in the original post that for purposes of this puzzle, Y is a vowel. So when it says "no other vowels," that includes Y's. For example, "a word with two A's and two I's, and no other vowels" should not contain any Ys at all.

In general, the words I like to work with are the standard "word game" variety. I avoid proper nouns, foreign words, slang, contractions, hyphenated words, or acronyms. I also try to avoid jargon.

However, there are some hyphenated words among my solutions to this puzzle.

Let us agree that shorter words make "nicer" solutions to a puzzle like this - so as a challenge, try for the shortest word you can find in each case.