bonanova

Three out of four. Close.

Noting that infinity is not an algebraic quantity like zero is, I'm wondering if this is a picture puzzle and not a math puzzle?

Treating the capital letters as geometric shapes, I classify them as "naughts" and "ones." The naughts are: ABEFGHKPQRSTXY The ones are: CDIJLMNOUVWZ Applying the same assignment rule, if you can discern it, which four of the digits 0123456789 are "ones"? What is the rule?

Yes. Triangles and squares are the only regular figures that can go inside themselves without overlap. Triangle add or subtract .6 of the starting area; squares add or subtract an amount equal to the total starting area. Placed out side, you get another square, circumscribed at 45 degrees; placed inside, the entire area disappears in fascinating fashion.

The only way is to add smaller versions of the original shape onto its sides. So you wouldn't add triangles to the sides of a pentagon, for example, or vice versa. And the rule for the triangles on triangles construction was to add similar shapes with 1/3 of the perimeter to the middle third of each side.

"But not" was meant to nudge your thinking away from geometry and just bisect "twice.' The actual phrase I had heard was "Once effective, twice overdone." which applies variously to different situations, The original clue of "half overdone" was flawed anyway, as I think about it now. According to that phrase, half overdone would be a clue for effective, not for once. Nice going.

Let's eliminate overlap, and we've exhausted triangles. You can only add or subtract 60% of their original area. So let's just keep using regular shapes, and push that as far as needed.

So lead can be ironic ... Hmmm, let's keep on and go for gold?

Hi James33, and welcome to the Den. Close. You calculated about .62 x2. It's closer to .69 x2, or exactly 1.6 times the original area. For the next bit Nice idea, but you get into a zero times infinity quandry. If x is set to 0, then everything collapses to a point, and you lose the infinite length. What you can say is that the area can be made arbitrarily small by letting x become arbitrarily small. But the area will remain nonzero. Correct about the figure described in the OP. Can you imagine a process similar to that described in the OP that gives zero area?

First two are correct. The third is close, and it fits, but it will lead to dead end with the following clues. Progress, tho: only 5 and 6 have not been identified.

It is possible to increase the area of a regular triangle by placing smaller regular triangles on the middle thirds of its three sides. By so doing, you obtain a six-pointed star. The process can continue indefinitely. At each step, a smaller regular triangle is placed on the middle third of all the line segmens on the perimeter of the figure obtained from the previous step. Sketching the shapes obtained for the first few steps of this process is an interesting way to spend a few moments. The perhaps surprising result is that this process converges to a fractal-like figure of infinite perimeter but of finite area. Can you determine the area limit? A more interesting question arises. Can some similar process converge to a fractal-like figure of infinite perimeter but of zero area?

Another?

Clue.

If I raise my right hand my mirror image raises his left hand. If I wink my left eye he winks his right eye. Why are top and bottom not reversed as well?

"Will" is, um, future tense. Which is why it made sense for the man to say "I'm innocent, and I'll never do it again!"

Ouch. That's 0 for 4. 2 isn't that hard if you think about it. But it is a bit surprising if you don't.

I'm guessing I don't have to worry that I'm doing your homework here

Good job. 7-13 are the words I had in mind. Clue 5 needs to be fixed. Unfortunately I had in mind a saying that I heard as a child and I erroneously thought was common parlance. I googled every possible reference to it and came up empty. It must have been colloquial to a very small region and then disused over time. So ... A new Clue 5: 5. Twice bisected, but not quartered. I will edit the OP to reflect the change.

I believe it's all new. And I'm thinking ... I'm thinking the number might be astronomical. It's arithmetic, geometric, factorial or exponential. My money is on one of the last two.

That indeed ties a bow on it.