Logic Puzzles
Jo(e) has posted a logic puzzle over at her place (July 10th’s post, “What Colour is Your Bandana”) that some of you might enjoy. Me? I hate logic puzzles. For many years, they made me feel more stupid than almost anything else I could attempt — right up there with playing chess. They were held up to me as the measure of the true intellectual, the kind of game that a really smart person could do in their sleep, the stuff that separated the merely well-read (me) from the genius. I got sick of the taunting — “c’mon, even my sixth-grade cousin can solve this!” and the implication that if I just worked a little harder, a bright person like me could surely work it out.

Well, you know what? Turned out that the big “clunk” that happens every time I try to do one of these “brainteasers” was telling me something. I have a very specific learning disability that has to do with doing spatial sequencing in my head. There are work-arounds for it (somewhat) using manipulatives but it really does make much of fancy-pantsy math a complete no-go for me. It has nothing to do with the amount of effort I put into it, the amount of time I invest, or my general intelligence.

So, if you’re a logic puzzle lover and you meet up with one of us who ask you for the answer, please don’t do the “but that would rob you of the reward of solving it yourself” thing. I am not encouraged when you tell me how easy it is and how readily I could do it if I just thought harder. Honest. I just want the answer in the hope of figuring out why my brain does and doesn’t work like yours.

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Bridgett, I figured it out. Do you want the solution?

Comment by nmJuly 11, 2007 @ 4:57 pmYes. Absolutely.

Comment by bridgettJuly 11, 2007 @ 7:03 pmI thought I had it figured out. But now I see that I didn’t. I saved half the gnomes for sure, but not 99%. So I don’t mind telling you my wrong solution, but when I get it all figured out right I’ll let you know. The last gnome in line (the first one to speak) says the color of the hat in front of him. If his hat is the same color, he lives, and if not, he has sacrificed himself for the greater gnomish good. So, obviously, the next gnome in line is safe: the gnome behind him has told him the color of his hat. Then the third gnome in line says the color of the fourth gnome’s hat, and so on. At least half of the gnomes make it in this way. I haven’t made it up to 99%, though. My bad, to have thought I did.

Comment by nmJuly 12, 2007 @ 7:56 amThanks anyhow. If you do get it, please share.

That’s the solution that my husband came up with as well. And I was happy as a clam and would have gladly accepted that as a right answer, until he thought some more and realized that this would net a 99 or more safety rating. Then he had to had to explain why this wouldn’t work, with playing cards, because I couldn’t visualize in my head why it was wrong.

Honestly. These things just make me feel completely lost. It’s like someone turned out the lights in that part of my brain. Lack of spatial sequencing is somewhat of a crap disability for a historian to have, but on the other hand, it has no effect (as far as I’ve noticed) on stuff like sequencing narrative or chronology. It made chaptering a nightmare, though, as I have to “see” everything in my head at once and keep it there while writing. (I am probably an article/essays person rather than a big book writer because of this.)

It also has some superficially annoying effects that show up when we’re doing home repair. I have problems keeping order of operations in my head — first do this, then we have to nail this to that — which I had chalked up to lack of experience until about the tenth time through doing this stuff. I’m not just “forgetful”; the information drains out of my head because there’s no place for it to stick. Now I just keep a list on hand to consult and everyone is much happier. I also have marked difficulty figuring out how to fit furniture in a room without drawing a diagram with measurements and stuff…once I see it on paper, then I can “see” it in my brain, but before that…nothing.

Like anything else, one learns to work with it.

Comment by bridgettJuly 12, 2007 @ 8:15 amI have a lot of similar problems, bridgett. I do mental math completely batty, spatial stuff is often out of my reach, I get lost really easily… and so on and so forth. I actually wind up doing big writing projects by cutting and pasting… literally. I write out all the things I want to quote, and then rearrange it on the floor… then I write bits to connect it, and cut that up, and rearrange it on the floor, and just keep going until I hit the time limit or it makes sense.

I got the same answer nm had, honestly… but she’s right; it only gives you 50%, not 99%.

So I googled it… and found something complicated with three colors, a somewhat convoluted two-color explanation in a combinatorics paper (which nonetheless refuses to give the answer to

that specific problem, despite giving the answers to the four similar problems considered), a thread about a similar problem.If I were any good at all with this kind of math, I could probably pick out the important bits and explain them… but I can’t. Symbol systems cause my brain to turn to mush. (I actually wrote my entire calculus AP, for instance, in the form of short-answer essays. I can’t do algebra worth anything, so I couldn’t solve the problems for values… but I knew what to do and I wrote that instead. “To solve this problem, you would take the derivative of this, which would use this formula and yield an answer in this form. You’d then take this bit of that answer, multiply it by this bit of this other thing, and then plug it into this formula. That would give you an answer in the form of this, which should tell you how fast the thing is going.”)

I hope someone else figures it out.

Comment by magniloquenceJuly 12, 2007 @ 3:33 pmSee post above for solution.

Comment by bridgettJuly 12, 2007 @ 10:12 pmOk, Mags says I have to take a shot here at explaining this. Here goes.

Before they are lined up, the Gnomes all agree that whoever is at the back of the line (Picked first) is pretty screwed. So that Gnome, they all agree, will call out a color to tell them how the rest of them look. If there is an even number of Gnomes wearing red hats, he will call out “Red!” and if there is an odd number of Gnomes wearing red hats, he will call out “Blue!” The Evil Dictator comes to the first Gnome (Who is standing at the back of the line). I will use the first picture on the blog as the example set, so you can reference it visually.

That first Gnome looks down the line and sees that there are four red hats. He calls out “Red!” and the Evil Dictator… Blows his little Gnomey brains out. Th next Gnome in line, wincing and shuddering, looks down the line and sees four red hats.

Now armed with the knowledge that there are, including himself, an even number of red hats, he calls out “Blue!” and… Survives.

The next Gnome in line looks down the line and sees three red hats. Confused for a moment, he then realizes that because the first Gnome saw an even number of hats and he sees an odd number (And the only other Gnome who could have been that last hat was not it), he must be wearing a red hat. So he calls out “Red!” and lives.

The fourth Gnome in line looks down the line and sees two red hats. Thinking about it, he realizes that since the first Gnome saw an even number of hats, he also sees an even number of hats and one red hat has already been picked out, he must also be wearing a red hat. He calls out “Red!” and lives.

Th fifth Gnome in line looks down the line and sees two red hats. Knowing there must be an even number of red hats and two Gnoms behind him are wearing red hats, he must be blue. He calls out “Blue!” and lives.

The sixth Gnome in line looks down the line and sees only one red hat left. Knowing that there must be an even number of red hats, he deduces that he must be wearing a red hat. So he calls out “Red!” and lives.

The seventh Gnome in line looks down the line and sees that same one red hat left. Knowing that there must be an even number of red hats and that there are three Gnomes in red hats behind him, he deduces that he must be wearing a blue hat. He calls out “Blue!” and lives.

The eighth Gnome in line looks down the line and sees… No more red hats! Uh-oh. That means he must be wearing a red hat, since there are an even number of red hats total and three Gnoms behind him are wearing red hats. He calls out “Red!” and lives.

The ninth Gnome in line looks at the back of the last Gnome and thinks hard. He doesn’t see any more red hats. There were an even number of red hats, and four other Gnomes called out “Red!” and lived so far… He must be wearing a blue hat. He calls out “Blue!” and lives.

The tenth Gnome in line sighs. He can’t see anyone else. But he knows there must be an even number of Gnomes wearing red hats in the line, including himself. He also knows four other Gnomes have called out “Red!” and lived. He must be wearing a blue hat. He calls out “Blue!” and lives.

So by possibly sacrificing himself, the first Gnome in line saves all the other Gnomes. If the first Gnome had been wearing a red hat, he would have lived as well. The important thing to remember here is that no matter what color hat the first Gnome is wearing, he doesn’t count when the other Gnomes are trying to figure out what color hat they have on, because no one can see what color hat he is wearing. He has a straight 50/50 chance to live no matter what.

Incidentally, six Gnomes were killed in the formulating of this explanation. Don’t tell PETG.

Comment by breviloquenceJuly 12, 2007 @ 11:54 pmHah. I was all confused, and then I realized that in the time it had taken him to figure it out and explain it to me, you’d gone and posted again and I hadn’t refreshed the window.

Weeeird.

Comment by magniloquenceJuly 13, 2007 @ 12:04 amThanks! I so appreciate the time you took to do this. I think your explanation is more readily understandable than the one I posted, so I’ll use it instead.

Comment by bridgettJuly 13, 2007 @ 8:26 amHey, you could have emailed me for the answer. But I see someone else has already explained it.

Comment by jo(e)July 13, 2007 @ 4:27 pm