Good for them. They've actually done it honestly and on a fine enough scale that you can see what's going on if you look hard enough. This particular puzzle gets fudged too often.

Posted January 27, 2003 1:55 PM by Mike Scott

I brought the images into Flash, sliced them up into their component shapes, and dragged them into each of the two configurations. Didn't help. I'm still stumped.

Posted January 27, 2003 3:00 PM by Greg van Eekhout

Thanks, Mike. You provided the clue that allowed me to figure it out. Clever.

Greg, read Mike's post. Look again.

Posted January 27, 2003 3:31 PM by Stefan Jones

I think I see, but I still don't see how it can be. Thy hypotenuse is a straight line from corner to opposite corner of a rectangle. It should therefore exactly bisect the rectangle in both configurations.

Posted January 27, 2003 3:45 PM by Xopher

Alas, a Blog had a link to it too. I hate this, because I can't for the life of me figure out the gimmick, and I know I'm good at math. It's demoralizing, damn it.

Posted January 27, 2003 3:47 PM by Chris Quinones

I noted the same visual discrepancy Mike saw, and then did some quick calculations to understand why. I wonder if Mike used similar methods.

Posted January 27, 2003 4:07 PM by Charles Odell

I solved the first half of the problem with simple geometry, and Patrick finished it up with a bit of reasoning and his magic ability to see whether lines are straight or not.

(I can't. When I was a kid I caught a flying piece of broken glass with my right cornea, and have had a shaky relationship with rectilinearity ever since.)

It was a beautiful moment when I finished my calculations and realized that both figures were impossible.

Posted January 27, 2003 4:29 PM by Teresa Nielsen Hayden

A hint (well, more of a prod) for anyone who's still stuck on this:

What are the tangents of the acute angles of the
red triangle? How about the greenish triangle?

Posted January 27, 2003 5:15 PM by Dan Blum

OK, now I really feel stupid. I can't remember what a tangent is (except something that winds up hijacking a thread), much less how to calculate it.

If the key is that the "hypotenuse" isn't a straight line at all, then that makes sense. It sure looks like the line is farther up on one figure than the other, but I don't see how that can be.

Posted January 27, 2003 5:30 PM by Xopher (apparently an innumerate moron)

I just posted an illustrated solution on my site. The answer's in the angles.

Posted January 27, 2003 6:35 PM by Joe Maller

If you're going to slicing up the images and dragging them around, you might try dragging the one configuration over the top of the other and seeing how they line up. I saw this very picture a couple of years ago, and that's how I solved it then. (Which irritated me, a little, as once I'd done that it was clear that I should have been able to solve it by simple observation.)

Posted January 27, 2003 7:27 PM by Tim May

Under normal circumstances, I would stand up for brute-force methods alongside pure reason, but these are not normal circumstances. As we are doubtless about to be told, the mere fact that one cannot readily determine where the extra square comes from does not mean it does not contain fissionable material.

Posted January 27, 2003 7:42 PM by John M. Ford

I can't remember what a tangent is (except something that winds up hijacking a thread), much less how to calculate it.

Well, you don't really need to know how to calculate a tangent. All that's necessary is to note that the triangles are similar, except they're not. That is, if everything were kosher the two triangles would have to be similar (it's easy to show the angles match), so their sides would have to be in the same ratio. However, the big one's sides on the grid are 8 and 3, whereas the little one's are 5 and 2. The rest of the proof has already been done by Joe there.

Posted January 27, 2003 9:12 PM by Dan Blum

This is a fantastically evil puzzle.

In return, I offer the 12 Ball problem, stated (badly) at:
http://www.bikwil.zip.com.au/Vintage04/Editorial.html

Hopefully this hasn't already been covered here.

Posted January 27, 2003 9:36 PM by Jon Mann

Ah, I see. Next time I'll have to be more precise with my slicing.

Posted January 27, 2003 9:42 PM by Greg van Eekhout

Lovely comments, except I must protest. The *real* answer is that *those aren't triangles.*

They have four sides, not three. Start with the top figure. Letter the topmost point A, the point below it (the 90 degree angle) as B, and the far left point C.

You're not done. AB is a line segment. BC is a line segment.

BC is *not*. It bends, ever so slightly. In fact, it's *two* line segements. Point D is where the Red and Green triangles meet, forming two more lines, DB and CD. (Or, if you prefer, BD and DC, and YKIOKBINMK. YMMV. HTH, HAND.)

So, you're not dealing with triangle ABC, you are dealing with quadrangle ABCD. Let's letter the lower figure the same way, with A' through D'. (Ignore the gap created by the white cutout.) The trickery is that you swear it's a triangle, and you know that if A=A', B=B', and C=C', and you know one of the angles (the right angle at B and B') then the two triangles must be identical and congruent.

Which is true. But, we don't have triangles ABC and A'B'C'. We have quadrangles ABCD and A'B'C'D' -- and that eqaulity trick above (the Side/Angle/Side proof) doesn't work on quadrangles.

So, you can't assume that the angle at D and D' is 180 degrees. It's not. D is about 179 degreees, and bows into the figure, and D' is about 181 degrees, and it bows out. Therefore, the area of ABCD is less than the area of A'B'C'D'. By design, that area is one unit square.

Posted January 27, 2003 10:08 PM by Erik V. Olson

"Under normal circumstances, I would stand up for brute-force methods alongside pure reason, but these are not normal circumstances. As we are doubtless about to be told, the mere fact that one cannot readily determine where the extra square comes from does not mean it does not contain fissionable material."

This from the author of "Heat of Fusion." Brrrrrrr.

Posted January 27, 2003 11:43 PM by Patrick Nielsen Hayden

Reading now. A sentence has exploded inside my mouth: When he had gone, I asked the nurses for something to eat, and they brought me some chicken stew. It tasted wonderful, even between bites of J's data strip.

Posted January 28, 2003 12:06 AM by Teresa Nielsen Hayden

And now I'm sitting here crying. Good story.

Posted January 28, 2003 12:14 AM by Teresa Nielsen Hayden

For those of us lacking in really complete visualizations of the Macro Cosmic All, might we be so fortunate as to be informed of the title of the story in question, and perhaps even where it might be found?

Posted January 28, 2003 12:20 AM by Graydon

Heat of Fusion, by John M. Ford, and the running heads on this xerox don't identify the magazine. Try the ISFDB.

Posted January 28, 2003 1:06 AM by Teresa Nielsen Hayden

Better to not check the ISFDB, since it is not currently able to provide info--see the latest news.

The (increasingly hard to use) Locus Index shows "Heat of Fusion" in Asimov's Sep. 84, and in Hartwell & Cramer's The Ascent of Wonder. There might be others reprintings, but I don't feel like hunting through all four annual updates.

Posted January 28, 2003 2:00 AM by Kevin J. Maroney

Thank you!

Next step, finding an instance.

Posted January 28, 2003 3:23 AM by Graydon

"Fusion" was reprinted in David Hartwell's ASCENT OF WONDER. (It's not in the NESFA collection, but will definitely be in the Tor collection, forthcoming, we hope.)

The interest is much appreciated, and I hope you all like the story, though please do not expect it to be about current affairs directly, any more than Stan Robinson's outstanding novel was.

Posted January 28, 2003 4:30 AM by John M. Ford

Why are we all _so_goldurn_agitated_ alluvasudden?

Posted January 28, 2003 4:34 AM by John M. Ford

Are we? Uh-oh.

Posted January 28, 2003 6:56 AM by Teresa Nielsen Hayden

...well, everything I was seeing (and typing) was italicized, but maybe that was an artifact of my hardware, which has been a touch odd lately, or possibly my slightly slaunchwise retinas. Or perhaps tnh fixed it.

Posted January 28, 2003 2:35 PM by John M. Ford

Oh, that. I fixed it. Kevin accidentally closed an italicization with /a instead of /i. We thus find out that format tags carry over into all subsequent comments.

May this power be used only for good.

Posted January 28, 2003 3:25 PM by Teresa Nielsen Hayden

Test, test.

Posted January 28, 2003 3:28 PM by Teresa Nielsen Hayden

Test.

Posted January 28, 2003 3:28 PM by Teresa Nielsen Hayden

Nope. Doesn't work with color. Just as well, really.

Posted January 28, 2003 3:29 PM by Teresa Nielsen Hayden

It does in Chimera.

Posted January 28, 2003 4:06 PM by Stephanie

... althouth this little feature might be fixed in MT 2.6. Anyone used the Sanitize plugin before?

Oh, and for my fellow Mozilla users: A closing font tag. Back to normal...

Posted January 28, 2003 5:58 PM by Stephanie