bonanova

Nice problem. I'll get it started by observing that no matter where the point in placed,

@CaptainEd Thanks. I read somewhere (probably in a Martin Gardner book) that seven was the commonly accepted result (as given by @Pickett); this eight solution is a relatively recent discovery and published only incidentally in regard to a separate problem. A third party then recognized its relevance here. Surprising, because results of this type are things we imagine were settled centuries ago. Happy to share it here.

BMAD, plasmid and Rocdocmac are seated in chairs and behind screens such that each can only see the heads of the other two. And, of course, not their own. bonanova blindfolds them and places a hat on each. The hats are drawn at random from a box containing (1) Red hat, (2) Yellow hats and (3) Blue hats. i.e, { R, Y, Y, B, B, B }. Blindfolds are then removed, and each of them writes and signs a truthful statement, as follows: BMAD: My hat is one of two colors plasmid: My hat is one of three colors Rocdocmac: My hat is one of three colors Thalia, who is in the next room and can't see any of the hat-wearers, but who originally placed the six hats into the box, is given the written statements and now calls out the correct color of each person's hat. What are they?

Doing the derivative correctly,

Assumptions do matter. So here's some guidance, or not. I tried to word the puzzle in a way that it could be reconstructed. Theoretically at least, a normal person could spend an afternoon shelling peas, with a normal person's productivity, put her normal work product into a bag, and then those peas could theoretically be counted, and a distribution made, or compared with various candidate distributions. Basically (and mainly because I'm not Bushindo) I don't know what it would be. Then, my fictitious little sister can be assumed to have a hand that has a holding capacity of one to two orders of magnitude smaller than the capacity of the bag. But whether she grabbed as many as possible, or not, is a random outcome. (Handful may not mean full hand. Some may have been a better word choice.) I have no idea what implications devolve from that. But one could do the experiment 1000 times, in principle, and look at the distributions. Or, the matter might be known. Just not by me. Then, jhawk's observation is valid. I hadn't intended it, but it's there in the OP so it should be taken into account. If it helps, I have no problem with saying the total capacity of the bag is 10,000 peas; it was somewhere between 1/2 and 2/3 full; my sister's hand can hold no more than about 1% of the peas in the bag, and she may or may not have taken her maximum capacity of peas. Who understands sisters, anyway? Finally, if there are plural considerations that lead to different answers, I believe one of them predominates.

You can multiply by adding, so you can make squares that way, too. 12 = 1 22 = 2+2 32 = 3+3+3 42 = 4+4+4+4 ... x2 = x+x+x+x+ ... +x (x times) The derivative is 2x = 1+1+1+1+ ... +1 (x times) = x 2 = 1 What's wrong here?

Kudos, and the coveted bonanova gold star, to Rocdocmac.

So, how can one square touch more than seven other non-touching squares? Clue

That's very cool. It got me thinking. The sphere encloses the greatest volume for a given surface area. I don't know if this makes sense to ask but I'll try, anyway. For what dimension of space is the volume to surface ratio of a unit sphere the greatest? For example, in 3-D it's r/3 = 1/3.

I spent the afternoon in the garden, picking and shelling peas, collecting them in a large bag. When I got home my little sister reached into the bag and pulled out a handful of peas. What is the probability that she pulled out an odd number of peas? less than 1/2 1/2 greater than 1/2

This puzzle is inspired by the seventh of Rocdocmac's Difficult Sequences: In n dimensions, where a point is an n-tuple of coordinate values { x1, x2, x3, ... , ,xn }, the unit sphere is the locus of points for which x12+ x22 + x32 + ... + xn2 = 1. In one dimension this is a pair of points that encloses a volume (length) of 2 and comprises a surface area of 0. In two dimensions it's a circle that encloses a volume (area) of pi and comprises a surface area (circumference) of 2 pi. So the enclosed volume and surface area both start out, at least, as increasing functions of n. What happens as n continues to increase? Puzzle: Is there a value of n for which the volume reaches a maximum? Bonus question: What about the surface area?

To clarify: Picket's configuration has yielded the best result so far, but it is possible to do better. Puzzle still unsolved.

I'll just point out the interesting fact that with a tetrahedron, saying "two sides white" exhausts all permutations, since all pairs of sides share an edge. But with the cube this is not the case. So, I'm wondering what "patterns" is intended to mean: combinations? or permutations?

Best answer so far. Can you do exactly one (1) better?. Touching refers to red squares, which do not touch: Every point in the plane belongs to at most one red square. Donald Cartmill: Can you do exactly one (1) better?.

More red squares can be overlapped.

Gray squares overlap red squares but the overlap need not involve a corner (e.g. aligned centers and rotated 45 degrees.) Overlap can be any positive amount, from full overlap to a small portion of a corner, but positive - not zero. And of course there is no Red/Red overlap; red squares do not touch.

The area of the triangle is 1. What is the area of the white portion?

I randomly drew some squares on a sheet of paper and colored them red. Then I drew a gray square of equal size and counted the number of red squares it touched. Not very many. I forget the actual number, might have been 4 or 5. But it made me wonder: What is the largest number of red squares that a single gray square can touch? The squares are all of equal size and none of the red squares touch each other.

Possibly ... ?

Looks like

You're right. I botched it. rodomac's wordless pics were in my mind, but they never made it to the keyboard.