bonanova

Try again. It seems to work now. Or tell me the post number and I'll do it.

Not for idealized basketballs. However, if we stop the bouncing at Plank's height, you could calculate the final color.

I've seen it also with allowable regions (if symmetry is present) for the endpoints of the short horizontal line.

Which rationals XOR with 0.01001010... to give e/3 and pi/4?

Nice. I was thinking of doing a Shakespeare series. Game?.

And after it stops bouncing?

Right. Column 2 is only a subset of the rationals.

Clue: RARVABOREIRLSEDOEUELESETS <> ROSES ARE RED, VIOLETS ARE BLUE. Puzzle: TINFLABTTULAHSORIOOASAWEIKOKNARGEKEDYEASTE <> ? ADKFATATIKAWENITHNHTNTIEORODEOLDCSUNCLKTHKSHST <> ? SICSBTWNSUSOWYHHEUCUEOOECCNESRCETTEESEEESDSD T <> ? THOBPSHAGOEULAAORILRERLUEVPATUREKSBSNERRIOUS <> ?

Nicely done k-man.

Zeno indeed is watching this ball. Not every infinite series converges, but the infinite sequence of bounce times in this case does sum to a finite number. According to Xavier, 9.31 seconds after the ball is dropped, it comes to rest. Part 1 of the basketball puzzle is solved. The OP has been edited to include Part 2.

By elastic I would mean returning to the full height h. This ball is not elastic. By ideal I mean to say neglect air resistance, energy loss to acoustic processes, energy lost to deformation of the ball that might depend on impact velocity, and so on. That is, this idealized ball has none of these bothersome second-order details; it has a behavior that is completely described for the purpose of the puzzle as "returning to half the height, on each bounce, from which it fell." Aside from this, the laws of physics, in particular a constant acceleration due to gravity, apply.

Cantor tells us that infinities come in "sizes" or cardinalities. Some infinities are "larger" than others. The smallest infinite set comprises the counting numbers, 1 2 3 4 5 ... and any other set that can be put in a 1-1 correspondence with them. Such infinities are called "countable." The rational numbers are also countable, but the real numbers are not. Occasionally someone will attempt to pair the counting number with the reals. One such scheme is shown below, and the argument goes as follows: The left column is the endless list of integers in numerical order The right column contains a decimal fraction formed by reversing the digits and placing a decimal point in front. Since the left hand list proceeds without limit it eventually contains every possible sequence of digits. Then the right hand list also catches every sequence of digits and thus represents the real numbers less than unity. Numbers greater than unity can be constructed by sets with a shift in the position of the decimal point. The union of two or more sets of equal cardinality has the same cardinality as its component sets. This correspondence is sufficient to prove the two sets have the same cardinality. The argument is obviously flawed, and the question is to expose the flaw. Here are the lists: Integers Decimal fractions 1 .1 2 .2 3 .3 4 .4 . . . . . . 10 .01 11 .11 12 .21 13 .31 . . . . . . 100 .001 101 .101 102 .201 103 .301 . . . . . . 1234 .4321 . . . . . .

A square may be dissected into any number n of acute triangles, provided that n is 8 or greater. Show such a dissection for n=8.

An equilateral triangle cannot be dissected into two new equilateral triangles, nor into three equilateral triangles. Is there a highest number of equilateral triangles into which a single equilateral triangle cannot be dissected?

An idealized basketball falling from a height h bounces from the floor to a height h/2. Tell us two things: The ball is dropped from a height of 1m. Does it come to rest (stop bouncing) in finite time? Xavier, in shows us that after 9.31 seconds, the ball comes to rest. We now specify that the ball is blue initially and on each bounce it changes color, alternating between blue and red. After the ball comes to rest, it ceases to change color. Question 2: what is its color after coming to rest?

It's the probability of picking the number 0.500... from all the real numbers in [0, 1].

Once you require a "perfect" shuffle, the only randomness is the choice of in-shuffle or out-shuffle (where top and bottom cards never move.) Every card moves to a known position, and all you do is calculate whether a particular permutation of the first 52 integers is random. I'm interested to think what happens when the shuffle itself is random -- anything from "perfect riffle" to cutting the deck at a random card, thus interleaving any number of cards from 1 to 51. A single shuffle could then put the bottom card on top, or vv. You could assign a distribution of probabilities governing what happens, but that is all.

I gave it a try. Too many options to sort out. That is to say I could not program it.

As noted by dm92 in post#3, statement [4] in the OP is loosely worded, in a way that leaves its meaning unclear. "Zob is one and a half times larger than one of the islands. meaning 150% larger than. i.e. 250 compared with 100. What is probably meant is 50% larger than, or half again as much. i.e. 150 compared with 100. In the early days of dual processors, manufacturers claimed performance increased by nearly 180%. They also, both multiplied and added. What was true was performance nearly doubled. What was (either ineptly or slyly) claimed was it nearly tripled. And it left one to wonder whether an increase of only 80% would imply a performance degradation.

Got it. Similar to

This is one of the best puzzles of this type I have seen. It has several levels, and it is as demanding as puzzles that have many more variables that are difficult only because they are big. chocchief, look forward to more if you have them.

It's intriguing but the idea isnt clear. Maybe if we saw one solved,

I have most of yours and I'll add the obvious 30: 10. But I don't have a clue regarding a formula.

I agree with this solution. Nice puzzle.