A hat room contains a wall with 49 pegs, arranged in a 7-by-7 square. The hat clerk has 20 hats that are to be hung on 20 different pegs. How many lines, containing four hats in a straight line, is it possible to produce? A line can go in any direction: horizontally, vertically or obliquely. To explain your answer, number the pegs in order, from 1 in the upper left corner to 49 in the lower right corner; list which pegs you put the 20 hats on, and give the total number of lines containing four hats in a row.I've got Ross puzzling over this with graph paper while I blog. I would point out that it's not entirely clear whether you are allowed to consider 30° and 60° angles in addition to 45° when he says "oblique." Opinions, anyone?

Assuming "oblique" simply means at any angle other than horizontal and vertical, Ross has an answer. My challenge will be to convey that pictorially on Thursday!

Based on how few people got last week's answer, I would suggest that this week might be a very good week to enter. Do that by sending your answer to NPR via this site right here.

Incidentally, before we get to this week's photo array, I want to share David's awesome comment about the six cities I culled from the American Idol tour: "If those are the cities that the winners go to, where do they send the losers?" And the answer is: Back home. (

*Hey, Dave -- if the implication is that those six cities are podunk places, you should know I've lived in two of them, and quite near a third. Just sayin'...*)

Hats! Twenty of them!! (I love it when picking photos is this easy.)

This last photo has a rather skimpy caption: Pav Hats 4. I understand they are meant to look like pavlovas, a divine dessert that's popular in New Zealand, where this photo was taken. But are they paper? Were they worn to a party or just displayed? We'll never know.

Time for ...

P I C K A R A N G E

This is where we ask you how many entries you think NPR will get for the challenge above. If you want to win, leave a comment with your guess for the range of entries NPR will receive. First come first served, so read existing comments before you guess. Ross and I guess last, just before we publish the Thursday post. After the Thursday post is up, the entries are closed. The winner gets a puzzle book of our unspecified choosing.

Only three hundred entries? Now I'm starting to worry that there's not even an NPR intern behind this, but rather some over-worked and under-paid mail clerk has been told to eyeball the number of emails! Needless to say, none of us picked that low. But a geometric puzzle? The number next Sunday could be even lower. We have prizes (I restocked recently), so it's totally cool if someone wins!

Here are the ranges:

Fewer than 50 50 - 100 100 - 150 150 - 200 200 - 250 250 - 300 300 - 350 350 - 400 400 - 450 450 - 500 500 - 550 550 - 600 600 - 650 650 - 700 700 - 750 750 - 800 800 - 850 850 - 900 900 - 950 950 - 1,000 | 1,000 - 1,050 1,050 - 1,100 1,100 - 1,150 1,150 - 1,200 1,200 - 1,250 1,250 - 1,300 1,300 - 1,350 1,350 - 1,400 1,400 - 1,450 1,450 - 1,500 1,500 - 1,550 1,550 - 1,600 1,600 - 1,650 1,650 - 1,700 1,700 - 1,750 1,750 - 1,800 1,800 - 1,850 1,850 - 1,900 1,900 - 1,950 1,950 - 2,000 | 2,000 - 2,050 2,050 - 2,100 2,100 - 2,150 2,150 - 2,200 2,200 - 2,250 2,250 - 2,300 2,300 - 2,350 2,350 - 2,400 2,400 - 2,450 2,450 - 2,500 2,500 - 2,750 2,750 - 3,000 3,000 - 3,250 3,250 - 3,500 3,500 - 4,000 4,000 - 4,500 4,500 - 5,000 More than 5,000 More than 5,000 and it sets a new record. |

Our tie-break rule:In the event that a single round number is announced, AND two separate people picked the ranges leading uptoand leading upfromthat round number, the prize will be awarded to whichever entrant had not already won a prize, or in the event that both entrants had won a prize already or neither had, then to the earlier of the two entries on the famous judicial principle of "First Come First Serve," (or in technical legal jargon, "You Snooze, You Lose").

## 24 comments:

I would think angles other than 45 (and 90 and 0) are acceptable. (I'd quibble that 30 and 60 are actually impossible and you actually meant 63.4 and 26.6 but ... well that'd be obnoxious).

Portland, Maine is a very very nice place. Haven't been to the others but ....

I know at least one person let Will know the last time he used a Sam Loyd puzzle that he spelled his name wrong.

I started work on this last evening and by this morning was really hoping the on-air promulgation would be clearer than the on-line one.

It wasn't and I think there are too many equivocalities to spend time working on it.

And there certainly is no way to predict the Range given the apparently random ones reported recently.

Could someone have stolen the American Girl book from my mailbox?

Woozy -- Gimme a break, so I rounded up/down. Pfft. :-)

Mendo Jim -- Your prizes went out last Monday. I mailed three other things on the same day. The one going to Oregon has arrived; the one to Washington state and one to upstate New York have not arrived yet. Thus, I think it's a bit early to worry about theft. But if it reassures you at all, there is a second copy of the American Girl puzzle book as a RESERVE!

Actually, both of us are assuming a regular square grid. The more sensisble, albeit uncommon, way to express lines would be to use inclination rather than angles. e.g. "2 to 1" rather than "60 degrees". This way we can see we can he lines of 1/7, 3/5, 7/2, etc. without having to guess at what their actual angles are (which, by the way, would be the arctangents of 1/2, 3/5, 7/2, etc.) Of course, none of *those* lines can have four points.

I would imagine this type of geometry puzzle will have the least number of correct answers, except as it is a Sam Loyd puzzle, I imagine the answer is googlable and will be.

As a simple four by 5 grid will have 13 lines, we can assume the answer will have more lines than that. I've got a clever one so far that has 16 lines but I figure I ought to be able to beat that.

As Mendo Jim implied, the statement of the puzzle is not completely clear. How many rows of 4 hats are in a line of 5? If there are hats in spaces 1, 2, 3, 4 and 5, are there only one row of 4, two rows of 4 (1234, 2345) or five rows of 4 (1234, 1235, 1245, 1345, 1345)? Using the latter definition (probably not what Will intended), a row of 7 hats will lead to 35 different rows of 4.

I still haven't decided on my answer, but I will guess 200 to 250 entrants.

The American Girl puzzle book arrived in good order and just in time for delivery to my Craigslist customer.

We plan a little trip to the Islands on the proceeds, but when we return I would like to talk to you about purchasing the "reserve" copy.

Thanks and cheers!

I think it is interesting to speculate on how big a "Whoops" Will will cop to on this week's challenge.

David -- I read the puzzle as specifying that you can only count those lines with precisely 4 hats in them. A line with 5 hats wouldn't count at all, just as a line with 2 or 3 hats doesn't count.

Mendo Jim -- C'mon, at least let everyone know you got a REAL puzzle book as well. :-)

I will take the 300-350 range. I think I have an answer this week. Last week, I did not.

Yep, the real prize was a quality book (bound and with a self-marker like American Library tomes).

A good little bunch of posers for when you want to feel stupid. It's "Almost Impossible/Word Puzzles" and I like it.

Just for fun, I'll pretend Will tells us how many folks actually got the hat answer he intended and guess fewer than 50.

You may NOT have more than 4 hats in any line.

Will should have stated this.

It makes no difference at all from which corner you begin numbering the squares in the grid or if you go clockwise or counter clockwise. There would be no way anyone could possibly tell.

So, Will put in a totally irrelevant item and then neglected to include a very important one. He did give a clue in not saying diagonal.

I'm glad tomorrow is Thursday and that I didn't spend any time working on this challenge.

I have to admit that I missed the clue in Will's saying oblique instead of diagonal. I think.

I'm pretty sure you can count lines with five hats as a line with four hats. But it only counts as one line. The puzzle *does* say how many "lines containing four hats" not "how many set of four". After all, you wouldn't count the line with #1,2,3,4 as *both* #1,2,3,4 *and* #4,3,2,1 so why would you count #1,2,3,4,5 as both 1,2,3,4 and 2,3,4,5? It's only *one* line.

It seems unlikely that the answer will have any 5 hat lines as that'd be inefficient. Probably.

As for the numbering, that was only so that the submitters can have a method of communication. So I don't think it's accurate to say "Will put in a totally irrelevant item and then neglected to include a very important one".

I don't see how one can know what the maximum number is. I submitted 16 and I have a hand-wavy nonconvincing argument as to why I think it is *likely* to be the max but I have no way of knowing whether it is the max or not.

I'm guessing 150 to 200.

woozy:

You may be "pretty sure," but I know that you may not have more than five hats in a line, row, column. Will did not state the puzzle properly. I am not assuming this. I know this for a fact. I cannot explain this now because I do not want to give the answer away. And it is completely irrelevant to give meaningless direction as to how to number the grid as it is impossible to tell from which corner another person started from or from which direction he went. All that needed to be said was to number the grid beginning at any corner. And even this is silly as any fool knows how to number a grid.

Also, you do not have the correct answer yet.

Good luck, if you believe in luck. I don't.

I've had "My Hat It Has Three Corners" going around and around in my head for most of the week. One way or another, I intend to exorcise it this afternoon.

Okay, skydiveboy, The challenge is up to you to show that have an answer with five hats in a line with *more* lines than the "real" answer.

As for the "completely irrelevant" and "meaningless direction", as it was *entirely* an instruction on how to submit the anwer and nothing whatsoever to do with *solving* the puzzle I don't see that your complaint is valid. Yes, which corner you start from and which direction you go is utterly arbitrary, but had Will not got specific instructions (even though they are arbitrary) people would not have known how to submit.

Maybe I see your point, Skydiveboy, in that the answer (if Ross's answer is correct) is utterly symetrical, you would get the same sequence whichever corner or direction you go.

But had Will Shortz actually *said* that it doesn't matter which corner you go then that would have been a *huge* hint that the answer is perfectly symmetrical.

Likewise, if you meant, that you can't have any line with more than four hats *because* the answer doesn't, then stating such is *not* nescessary to the puzzle and gives away another big hint.

For a four hat/line maximum to be a *required* instruction, would mean that if you had left it out, it'd be possible to have a graph with a 5 hat line that has *more* lines. Which, maybe it is, but I doubt it.

(Again, I don't actually see how one can ever know if they've got the maximum number of lines. I'm pretty sure Ross's answer is the most but I don't see how one can *know* that.)

Woozy:

Your post makes no sense to me.

I never said I had a solution that had more than four hats or more than 18 lines. I said it was not allowed in the original puzzle. I suggest you read my posts with more care.

"Arbitrary" is not the issue. The issue is that it makes no difference which corner you number from. It was a complete waste of space to specify something that cannot even be ascertained from the submissions. However, the question of having more than 4 hats in a row caused much discussion.

I do have a solution with four rows of 4 hats each that equals 18, but I am not going to go to the trouble of describing it to you. I am reasonably sure Will received a few submissions with this or similar solutions which were specifically forbidden in the original puzzle. I hope this satisfies you, but I somehow doubt it will.

Woozy,

You just don't seem to see what is so obvious to me. It makes no difference from which corner or which direction even if the puzzle answer is not symmetrical. Think about it enough and you should see this. This is simple logic.

"This is simple logic. "

No, of course it makes no difference what corner you start in nor what direction you go in. But as those "rules" were about how to submit an answer and had *nothing* to do with the actual puzzle I see nothing wrong with Will's arbitrary and "unnescessary" instruction. Especially had he *not* indicated what corner and what direction to go, thousands of people would have been confused.

"I said it was not allowed in the original puzzle."

But if that rule isn't *nescessary* then the puzzle has actually been *improved* by Will leaving it out. (Just as rules stating "You must hang a hat on each of the four corners" or "You can't have a line with a 4-1 slope" are unnescessary and make the puzzle worse.) Because having five hats in a line creates a solution with fewer than 18 lines, such a solution is simply *wrong* and there is no reason for the rule to be explicitely stated.

(Sorry to everyone else)

>>I suggest you read my posts with more care.

Your post says precisely:

"You may NOT have more than 4 hats in any line.

Will should have stated this."

No, he shouldn't have. As the solution is the same whether this is stated or not, it doesn't matter whether he stated it or not. In fact, it's better without it as it makes the puzzle more flexible.

"It makes no difference at all from which corner you begin numbering the squares in the grid or if you go clockwise or counter clockwise. There would be no way anyone could possibly tell."

True but so what? This isn't a rule for the puzzle but how to write the answer. Had Will said simple "Start at a corner and number from 1 to 49" many of the players would have asked "What corner should I start at? What direction do I number in?" without realizing that it doesn't actually matter. Thus Will was making it easier on everyone.

"So, Will put in a totally irrelevant item and then neglected to include a very important one."

So, I disagree with both your points. The irrelevent item clarified and simplified potential confussion and thus was not *totally* irrelevant. The fact that you can not have more than four hats in a line was absolutely unnescessary.

Hey wooooooozy,

Have you ever heard of spell check or proofreading?

Why? What issues do you have with my posts? I'm sure there are typos as I'm a bad speller and a careless typer but what issues of proof reading do you have?

But for your convinience, here or corrections:

June 12:

sensisble (should be sensible)

googlable (intentionally not a word but I thought it would be clear from context that I meant "capable of finding via a google search")

June 16:

nonconvincing (should be non-convincing)

June 17 8:55

anwer (should be answer)

9:17

symetrical (should be symmetrical)

nescessary (should be necessary, I actually do this about four or so more times. tsk.)

10:13

explicitely (should be explicitly; that *is* a bad one)

10:28

irrelevent (should be irrelevant)

confussion (oh, this one could be funny if put it the right hands. "Confussion: The state of being so befuddled that one actual suffers a head injury.")

Those are my spelling errors. Is there anything else you find objectionable or unclear?

Speaking of proof-reading, what exactly did you mean by "clockwise" and "counter-clockwise"? I chose not to comment on those because I didn't feel pointing them out would be relevant.

Well waddya know.

In restating the puzzle, Will Shortz did this week state that the lines must contain exactly four hats, a fact he had indeed left out the week before.

Nonetheless, I still maintain such a statement, as it doesn't alter or affect the solution it wasn't "very important".

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