Ever obsess on something trivial when you know you have a ton of work due about an hour from yesterday? Here’s my obsession for today:

In the last Math Mojo Monthly Newsletter (look up at the top in the left menu bar of this post to make sure you get it. It’s free, of course) I posed this puzzle:

What is pizza = 307.72” ?

I just wrote the answer for this week’s edition of the Math Mojo Monthly (hurry and sign up for it now, so you’ll get the issue when it comes out, or else you’ll miss it – nudge, nudge!)

I got some interesting answers back, but the two most representative ones were:

The circumference of a circle whose diameter is 49 ” will be 307.72″, taking pi=3.14 So that is the size of the pizza — a 49″ pizza !!!

To which I wrote back:

Good try. How do you figure, though? Pi times diameter wouldn’t give you 307.72, would it?

Hint: Remember, this is a puzzle. There is a puzzle element at work here. What would make the question clever?

The reply came:

Oops. I meant to write 307.72/(2pi) = 49. So Area = 2pi*r² means a 7″ pizza. That’s your pizza.

Which was the same as the other representative answer, namely:

307.72 is the surface area in square inches of a pizza or circle with a radius of 7″

To which I replied:

Keep trying. You’re not too far off. Once you get it, you may get the “aha!” moment that the puzzle is actually after.

Both were noble attempts, but a little checking would have shown the errors. Plug those figures into the formula for a diameter of a circle, and you won’t get 302.72 . Each of those pizzas would have been pretty far off from the diameter of a standard pie, though. That could also have been a clue that they were off.

Here’s the reason I wanted to write this post,though. The first set of answers came from a mathematician. This gentleman is leaps and bounds ahead of me in his understanding of mathematics (I’m not a mathematician, just a math dilettante). He got the answer wrong. My point is: So, big deal? He was also the first one to write in. He had the interest and the effort going for him.

Too often schools surpress interest and discourage effort by their rush to judge students by testing them. It’s that false “accountability” platitude.

So I wrote back to the gentleman and we had a nice dialogue. I hope he communicates on this blog often, because his insights would certainly benefit readers.

This is what I wrote back to him, concerning his answers. I think it may be motivating and encouraging to some people, so I’m reproducing it here:

One of the things I am most concerned with about math are common mistakes, and why people make them.

Too often students feel intimidated about making mistakes. I think standardized testing is what kills this spontaneity and curiosity. I could be wrong, but I think that’s the way to bet.

One of the missions of Math Mojo is to relieve some of the fears and stress of students.

I think a great way to do that is to show them that anyone can make simple mistakes, and that there is no shame in that. The shame is simply not trying. There is no shame in trying to making honest mistakes.

I am frequently a great example of this. I make plenty of mistakes on my website. It has a great advantage for me, which I did not expect. Every time I make a mistake, I can count on more comments to that blog post than usual! I don’t have to make the mistakes on purpose — because I make enough without trying.

A problem I have when posting puzzles, is that many people, especially young people, assume that I know what I am talking about, and I am the guy who makes up the puzzles, and I am some kind of genius. I think they assumed that I could solve these puzzles each time I see them.

Nothing could be further from the truth. I found a version of this puzzle on the Internet. The answer was right there with the question, so I didn’t actually have to solve it. I just found it very clever and interesting.

To tell you the truth, I don’t think I could have solved it myself. But I certainly would have tried, and probably wasted a whole weekend doing it. Of course it wouldn’t really have been wasted, it would have been fun, and I’m sure I would have learned many things along the way.

Which gets me to the point of why I’m telling you this: I think you may have made another honest mistake.

As you know, the formula for the circumference of a circle is pi times diameter. It could also be pi times two times the radius (because twice the radius is the same as the diameter). C = pi*d, or  C = pi*2r.

The formula or the area of a circle is pi times the radius squared. A = pi * r².

I think the confusion stems from the fact that 2r looks like r². But they are not the same. One is two times the radius, and the other is the square of the radius.

In second answer, you  said, “ Area = 2pi*r²“.  There is an extra, unneccesary 2 in that equation. I think the extra 2 might be due to the above explanation.

It is a totally common mistake, that even a mathematician can occasionally make. I would like to let my readership know that there is no reason to feel that someone must be “bad at math” to make such mistakes. Clearly, if a mathematician can have a foggy day, then so can others. It doesn’t mean you’re “bad at math.”

When I explain the solution to the puzzle in the next Monthly, I think you will find it very entertaining. I’ll bet you will try it on some of your mathematician friends in the future.

I really appreciate our dialogue, and I thank you for playing the puzzle in the Math Mojo Monthly. I hope you keep working on it, because I really appreciate talking with an actual mathematician occasionally. It keeps me on my toes.

This all brings me back, once again, to the value of puzzles. Puzzles are a great place to be able to make mistakes. You don’t get graded on them (unless you have a teacher who’s making a great pedagogical mistake). Puzzles are a safe place to practice and  improve your intellectual ability.

Another benefit of puzzles is that they are not always straightforward. They test and hone your ability to think laterally, creatively. This wasn’t just a simple equation. It was a challenge to think differently. It could have been solved without doing any equations at all if you thought like a puzzler instead of a mathematician.

The mathematician might only be looking at the numbers, where a puzzler may have been looking from a wider perspective, and seen the letters, and recognized right away that the “pi” in “pizza” might stand for π

Maybe I’m giving away too much here, but you might think about what the double z could stand for. And don’t forget the a.

I won’t be giving away prizes for anyone who solves this (I offered one in the last Chronicles, but nobody won). But if you have an complete answer, send it in before the end of the weekend and I’ll give you an “honorable mention” in the upcoming issue of  The Math Mojo Monthly newsletter. (Wow, it just doesn’t get any better than that!)

You may also be aware that the Friday and Saturday NYT Crossword puzzles are harder than the Sunday puzzle. Sometimes much harder.

I don’t know if I could solve many of them alone, but together, we are a pretty good team. Mimi can brainstorm and come up with things I could never get, and I can help filter out some of the wild Ideas she comes up with that would jam the puzzle.

By “jam,” I mean, if you put in an answer that turns out to be wrong, it will mislead you from getting the answers that cross that answer in the puzzle. It can send you down wrong paths, and keep you from noticing the right ones.

One thing that we’ve found on many puzzles, at least over the last few years since we started doing them, is that there will often be a mention of something that appears in the NYT that day. Often it’s from the magazine section.

Unfortunately, we can’t get delivery of the NYT where we live (too rural). I used to get the paper from the truck stop each Sunday, but I stopped a few years ago. So we don’t get a lot of the references anymore. (Stay with me – this will be relevant soon.)

We usually do the puzzle during meals, and we get Sunday in one meal, on average, and Friday or Saturday usually take two or more meals. That leaves lots of meals during the rest of the week without puzzles, so we go back into the NYT online puzzle archives when we run out of puzzles. We subscribe to the puzzle section of the NYT online. It’s about $39 a year, and totally worth it.

This week we were doing the puzzle for Friday, Nov. 29, 1996 (from the archives). We almost finished it by the third meal. But there was one particular spot that was getting us. 24 down was: “Heraldic bands.” We have no Idea what this could be.

We had all the letters for it except for two. The first was the second from the top, which crosses 28 across: “U.S.D.A power agcy.” We originally thought it must be the F.D.A, but the second letter in it is definitely “e.”

The other missing letter was the second from the last letter, which crosses 39 across: “Els with tees.” We had all of the other letters in 39 across. “E ,R, N, I, ___. Yes, I know some of you know this right away, out of context. But we were totally lost. We know nothing about the context. Of course we could guess at “Ernie” and we imagined it would be right. But we had no corroboration.

Back to the point I made above, about how sometimes you get hints from what’s in the newspapers that day. This is a very cool thing the NYT does. It reinforces what you’ve read that day. And this is one of the things that I believe puzzles help your mind with. Any time you cross-reference information in your mind, you are building connections. These are sometimes referred to as “neural pathways.” You can read more about such things here:

• neural pathways: http://www.neuralpathways.org.uk/articles/repetition.htm

• neuroplasticity: http://en.wikipedia.org/wiki/Neuroplasticity

(Please keep in mind that I am not an expert in this field, and you should take these thoughts and websites as a “jumping-off point” and not as Gospel.)

As we did the Friday, Nov. 29, 1996 puzzle, we figured we could get no help from the current newspaper. We had started the puzzle two days ago.

So how surprised was I when my wife brought home the paper from the nearest “city” (population: about 15,000, Motto: “City of Pizza and Beer – We Used to be Somebody, but then they took the Trains Out”) and there was an article in the sports section which gave us a hint? We normally don’t read the sports section – but Mimi scanned the caption of the photo on the cover. The picture was from this week’s U.S. Open at  Pebble Beach. There were two golfers kneeling and checking out a shot.

Amazingly, the caption started, “Tiger woods and Ernie Els Line up putts on the 12th hole…”

“Els with tees.” Ernie Els, the golfer! Get it?

Here’s the thing – If we’d have simply gone with our “instincts” and put in Ernie to begin with, we would have forgotten about it later, because we would have given ourselves the feeling that we had “solved” it. We may have gotten the right answer, but it would have been a dumb guess, and we would have learned nothing.

By not taking the easy way out, and staying curious, we learned something. This was a shining example of how that works.

Some people may think, “Hey, big deal – so you learned something about a sport you don’t care about.” But they’d be missing the point. We are developing a healthy habit. By not being mentally lazy, and keeping our curiosity open instead of giving ourselves credit for “solving” something without understanding it, we are creating more neural pathways. Not just this time – it is a habit.

Doing puzzles is a perfect way to develop such habits. Think of it as like playing a sport. There really is no sense in hitting a little dimpled ball around with a crooked stick. But you’d be developing patience, hand-eye coordination, and the ability to handle hours and hours of boring nonsense. (Whoops… sorry golf fans.)

As much as I like puzzles, I’ve come to think of math as about the ultimate puzzle. Not in the sense of it being confusing (because by its nature it aims at reducing confusion and creating clarity), but in the sense that it has logical rules but requires creativity to understand and use in any meaningful sense .

If you like to sharpen your mind, make yourself more mentally resilient, learn something useful, and have a good time doing it, I don’t think you can do better than learn math and mathematical philosophy.

P.S. -  After getting Ernie there was one space to go. We were stumped. We’d have to guess. So we did. In other words, we didn’t fully solve this puzzle. We had to “cheat” and check the answers on the puzzle we’d downloaded.

What’s an orle? Check out this picture. The blue border is the orle. I told  you Friday puzzles could be hard!

Did I say hard? How about diabolical? Although REA is the answer to “USDA power agcy.”  I still had to do some serious Googling to find out what that meant. it turns out that it’s the Rural Electrification Administration one of the New Deal agencies created under President Franklin Delano Roosevelt, which was the foreunner to the Rural Utilities Service.

I also checked out Ernie Els. They guy was competing fourteen years ago when this puzzle was made. Now he is tied for second place in the U.S. Open. Pretty impressive. Maybe more impressive is that he has established the charitable organization, Els for Autism Foundation.

P.P.S. - There is another Math Mojo Chronicles post along these lines at Crossword Puzzle Digression. It mentions a good resource to help you understand the logic of the answers of the NYT crossword puzzle. Unfortunately that resource did not exist in 1996, so I was reduced to googling this time .

 ”What word best completes the following series:

ball, beach, circle, ______, eucharist.

The answers were (not exactly, but something along the lines of):

a) bottle

b) dusk

c) dough

d) summer

There are many things to be aware of when doing these kinds of “tests.” First is that they are a scam to get you to give them your cell phone number, so they can then spam you with tons of crap you are not interested in. 

Another is that there are many possible ways to look at things, and nobody has them all covered. Someone may discover a much more interesting algorithm than the one they want you to see. 

All that being said, what they are probably looking for is “dough.” 

Why? 

First I thought they were going for the amount of letters in each term, which would logically suggest that a seven-letter answer be the right one. But there are no seven-letter answers. Bummer. 

Then I thought I’d go with the fact that they are alphabetized, and a d-word would fit right in. But there are two d-words. So then I thought maybe it had something to do with the length of the d-words. Nah. 

Notice something else about all the words? Eventually I noticed that the second letter of each word was a vowel, and they were in a, e, i, _, u order (in other words, alphabetized.) The missing vowel was “o.”

That’s why I think the answer is “dough.” 

I didn’t make up the question, so don’t hold me to this. These “IQ tests” have no value, except to make you think (which isn’t so bad). But they hold no indication of your intelligence whatsoever, if you define “intelligence” as something more than the ability to waste your time seeing if you know some trivial baloney. 

If the ability to waste your time seeing if you know some trivial baloney really meant something, I’d be freakin’ Einstein. Nah, it doesn’t mean anything. But sometimes it’s fun.

When I was a kid, my father introduced me to the New York Times Crossword Puzzle. Dad commuted from our house to New York City every weekday, and often rode the Long Island RailRoad. He took the puzzle every day. He solved it in pen. He also solved the cryptograms in pen. I was impressed.

He taught me some basic logic for the solution of both puzzles. I did the cryptograms for awhile, but didn’t get into the crossword puzzle until a few years ago. My wife and I try to do it every Thursday through Sunday.

You may not know this (I didn’t, until my dad tipped me off a few years ago)….

…but the NYT puzzle gets progressively harder from Mon. to Sat. then gets a little easier on Sunday. Friday and Saturday puzzles tend to be brutal. Sunday is challenging, but not so hard that you can’t relax with it.

My team-mate (my wife) and I had been stumped by a corner of the puzzle from last Friday. It was driving us nuts (who the heck is Al Leitner?). Anyway, we usually get ‘em all within a day or two, but this one was tough, so I hit the web and googled some clues. I don’t like to cheat (OK, I do, but my wife has a little more backbone than I, and she usually won’t let me look up an answer until we’ve given it a few days).

When I googled the clue “boho-chic footwear” (something I feel a certain amount of pride in not knowing), the first site listed was “Rex Parker does the NYT Crossword Puzzle.”

Rex is a puzzle-meister, who keeps a blog of exactly how he solved each clue of each NYT puzzle, a day after the puzzle comes out.

What’s cool about it, is even when you normally “cheat” and look up answers, you often don’t know why the answer is the answer (“Ugg boots?” sheesh!) But Rex gives you his reasoning behind how he came up with the answers, along with plenty of general insights into the puzzle, and some witty banter. So instead of simply cheating (googling, looking up the answers the next day in the paper, etc.), and just “getting the answer,” with Rex, you actually learn something (imagine that!)

From now on, Rex’s site will be my “go-to” site when we’re stumped. I’ll try to resist as much as I can, although it’ll be tempting to go there often.

He’s also got a noir/cool site about Vintage Paperbacks and Other Cultural Detritus at Pop Sensation

Here’s a hint about the NYT puzzle. Although it’s fun to do in the paper, we live in a rural area of NY where there’s no home-delivery of the the paper. I got tired of going into the local truckstop (three miles away) four times a week (especially on winter days) just to pay an inflated, upstate price for the Times. So we subscribe to it online now. Not the whole paper, just the puzzle. It’s only $39.95 a year.

Here’s a cool benefit of subscribing. Not only does your subscription come with Across Lite (a program you use to do the puzzle with on your computer) but you can print it out, too. We print it out on two pages, effectively making the puzzle about four to eight times larger than the one in the paper. At my age (born circa 400 a.c.e.) that is easier on the eyes, and easier to do with a partner (even if she is much younger).

If you have children, the Monday through Wednesday New York Times crossword puzzle is a good way to get them started on a life-long, fun mind-exercise.

’nuff said,

Professor Homunculus

By the way, on Friday, June 18, 2010, I added this post about How Puzzles may Improve your Mind, which has some points about how puzzles help develop neural pathways.

Which is worth more, a pound of $5 gold coins, or a pound of $10 gold coins? Explain why.

Before I go into the reader’s observations, let me tell you the answer:

Since gold coins (if they are gold coins, and not some hybrid – and the question did say “gold coins”) are valued by their weight, it stands to reason that a $5 gold coin is worth half of a $10 gold coin.

To make up a pound of $5 gold coins, you would therefore need twice as many as you would to make up a pound of $10 gold coins.

As you can see, twice as many $5 gold coins as $10 gold coins will have the same value. Conversely, that means that half as many $10 gold coins as $5 gold coins will have the same value. In other words, a pound of $10 gold coins has the same value as a pound of $5 gold coins.

So the answer is: “They are worth the same.”

Now back to our reader – he wrote the following:

The fly in the ointment of this question is, “Who minted the coins, and when?”

In the very early days, the government minted the coins and would put $5.00 of gold into the $5.00 gold piece, and a $10.00 gold piece was double the weight. In this scenario, the pound of $5.00 coins might hold 16/1-ounce coins worth $80 total. And the 1/2 pound of $10.00 coins would hold 4/2-ounce coins worth $40.

Simple. But give it time, and the demand for the smaller denomination is great so the government modernizes the $5.00 coins (or worse, privatizes them), so they turn out to have only a gold plating and are made of nickel.

Then the Republicans come along and crash the money markets like everything else: low and behold, the value
of the bags changes in this way:

Let’s define your terms, Brian!

Yours truly,

Pierce Brosnan (Not real name).

My reply:

God, Pierce, I thought I was picky!

OK, to be really, really picky, I DID define my terms. I said “gold coins.” I didn’t say, “gold-style” (as in Kosher-style”), I didn’t say “virtual gold” (as in “virtual reality”), nor did I say “near-gold” (as in “near beer”), nor “faux-gold” (as in Faux News).

I did not call it “trompe le’oiel”
I did not call it “virtual.”
I did not call it “counterfeit”
I did not say that, not one bit.

I did not say that it was “pseudo”
Nor imply it was E.U. dough.
No implication of being “gilded”
“Gold coin” is clearly how I billed it.

I didn’t say ’twas from Franklin Mint.
(Had I done that I would have sinn’t.)
Lest your doubt not be abated,
Rest assured it’s not “gold-plated.”

I simply said it was of gold.
As gold, not “costume,” bought and sold.

I did not say ’twas “nickel-core”
Gold is it’s only stock and store.
It’s just as gold as gold of yore.
Now let the doubt be nevermore.

By the way, would it be OK to use this repartee in the next installment of the “Math Mojo Chronicles”? (I’d change your name. Would “Pierce Brosnan” work for you?)

Hotcha!

Professor Homunculus

The upshot on this whole exchange, is that you must assume that the coins are made of gold for the puzzle to work.

And that is a fair assumption, as my reply explains.
Addendum:

In Feb. 2009, Harvey Griffin made this observation, by Twitter:
The problem suffers from issue that $ gold coins are no longer minted. Value not based on weight, but rarity.

Professor Homunculus sez:

How right Harry is. And that is a good example of how, when you are doing “math,” sometimes we can’t see the forest for the trees. We get stuck on the equation that we want to solve, without realizing that there is more to the problem than fits in the “standard” equation.

P.S.  - This puzzle is also often phrased, “Which is worth more, a pound of $5 gold coins, or a half-pound of $10 gold coins?” It’s not exactly the same question, but it can be solved with the same logic. I think this phrasing may make it an even better puzzle. What do  you think?