bonanova

A square sits on the diameter of a circle with one corner touching it, as shown. Given the area A of the square and the distance x from its touching edge to the far end of the diameter, find the circle's radius.

Here is a simple puzzle using bullets fired at random speeds along a straight line. Every second, a gun shoots a bullet along a straight line.Each bullet has a random speed between 0 and 1. Bullets do not slow down; but if two bullets collide, both of them are annihilated.The gun stops shooting after 10 bullets have been fired.What is the probability that eventually all the bullets will be annihilated? Edit: BMAD previously asked if the bullets never stop firing, whether (at least) one bullet survive forever. It inspired a lot of debate, I think without resolution. This puzzle I think has a provable answer.

I'm going to mark this puzzle closed, with plasmid's answer as being the best statement of how to get to a proof. I may be able to resurrect my simulation computer and do a long simulation at some point. If that's successful, I'll post the result here. Best estimate, from the Queen of Spades result of a few years ago is 86% of the highest ranking card showing up in the left pile.

Great observation, and you'd only need two of them actually. Hidden Content Hidden Content Can you give a set of rational coordinates?

No. And you're right... But I didn't see how to move it there once I created it. Still learning the new site features.

this doesnt appear to be infinite though I agree with Pickett you just need a stronger magnifying glass.

Great observation, and you'd only need two of them actually.

Amid a flood of contradictory comments about how good/bad Windows 10 is, I installed it on my notebook last week, In general I like it and don't see anything buggy or undesirable. What is your experience? Thumbs up or down?

As promised ...

Is it possible to put an equilateral triangle onto a square grid so that all the vertices are in corners?

I'm surprised the result is as high as that. A simple game is to use a deck of 52 cards and ordering A 2 3 4 5 6 7 8 9 10 J Q K and C D H S in ascending order the expected highest raking card in the low pile becomes the Club Queen. That's the 45th highest card out of 52.

These two are too long to solve as anagrams, but they can be solved by thinking about what quotations they are describing, then checking to see that they are correct. Both I think are quite remarkable.

Dirty Room PickettA Rope Ends It PickettHere Come Dots RainmanCash Lost in 'em PickettIs No Amity PickettAlas! No More Z'sGenuine Class PickettIs No Meal PickettLarge Picture Halls, I betI'm a Dot in Place PickettThat Queer Shake PickettTwelve plus one PickettAccord not in it PickettTwo left to solve.

Here's the clue, then.

Let's finish this puzzle.

The solution requires only two steps. I've alluded to the probability that the first step involves using the compass. The puzzle has an interesting background: It is one of a group of problems that, for discriminatory reasons, was crafted to be very difficult to solve, but that nevertheless had a simple solution. In that sense it is an Aha! type of a problem. It is not at all clear how to proceed; but the solution, once seen, is, like, oh yeah ... but I never would have thought of it. Since that's probably the case now, I will not give the answer but I will allude to the first step, leaving out some needed information. If you read the clue, the problem will be simply to figure out the needed information and use it. The second step will then be easily found. OK, actually, the first statement in this post might now be enough of a first clue. The clue described in the paragraph will follow, if needed. I will of course award the coveted bonanova Gold Star for a solution.

If it's not obvious why the isosceles case fails ...

[1] Yes, I think so. Dissection lines must join an existing vertex to an existing side at a right angle. The process can be continued an infinite number of times, because a self-similar figures is reached after each group ot two lines. This says the first two cuts determine the size ratios, and those ratios remain fixed. Simple inspection will show the ratios are determined by the size of the acute angle.. I don't know of a dissection scheme for non-right triangles. BTW, and I'm sure this is obvious, I only had to mark ONE of the acute angles as being preserved, since the other acute angle is its complement. [2] There are some anomalies in the new version. I'm discussing this one with Site Admin at present. It happened in other threads as well. Meantime check the post dates to be determine the right sequence. To doubt, use the "Quote" feature, as I did here.

I share a raver's question about choosing a triangle to dissect. I think only certain triangles are amenable.

can you do it using only geometry?

A great engineering solution to the puzzle. Thanks. Suppose the ruler and compass were exact and precise instruments. Could you then construct the triangle exactly, on the first try?

I think this does it.

araver beat me to it.

What are you able to do with a compass? How might that be of use here?