BrainDen.com - Brain Teasers # bonanova

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## Everything posted by bonanova

1. Kudos, and the coveted bonanova gold star, to Rocdocmac.
2. So, how can one square touch more than seven other non-touching squares? Clue
3. 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.
4. 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
5. 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?
6. To clarify: Picket's configuration has yielded the best result so far, but it is possible to do better. Puzzle still unsolved.
7. 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?
8. 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?.
9. More red squares can be overlapped.
10. 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.
11. The area of the triangle is 1. What is the area of the white portion?
12. 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.
13. You're right. I botched it. rodomac's wordless pics were in my mind, but they never made it to the keyboard.
14. Maybe, maybe not.
15. I, as well, stand by my answer. Caveat:
16. To clarify. You don't have any of these abilities: Make a line segment of equal length to another segment, or equivalently make equally-spaced pairs of points. (You don't have a pair of dividers.) Make a line segment that is parallel to another line (segment). Make an arc. (You don't have a compass.) But you do have the equivalent of a straight-edge.
17. Don't see how it matters, unless ...
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