superprismatic

Two players play a simple game called Odd Todd. They start with a pile of an odd number of stones. Players take turns alternately. On his turn, a player may remove one or two stones from the pile. The player who ends up having taken an odd number of stones wins. Under what conditions can the first player force a win? When can the second player force a win? What are the winning strategies in both cases?

Let H be the number of times heads comes up in 1,000,000 flips of a fair coin. Which is more likely, H = 500,000 or H ≤ 495,000 ?

Here's a 4×5 grid made up of squares: -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- | | | | | | -------------------------- [/code] And here are 5 shapes, each made out of 4 squares: [code] Shape A: ------ | | ---------------- | | | | ---------------- Shape B: ----------- | | | ---------------- | | | ----------- Shape C: --------------------- | | | | | --------------------- Shape D: ------ | | ---------------- | | | | ---------------- Shape E: ----------- | | | ----------- | | | ----------- Can you fill the 4×5 grid with these five shapes? You are allowed to flip them and/or rotate any or all of them.

Consider a list of all permutations of the alphabet put in alphabetical order. This list looks like this: ABCDEFGHIJKLMNOPQRSTUVWXYZ ABCDEFGHIJKLMNOPQRSTUVWXZY ABCDEFGHIJKLMNOPQRSTUVWYXZ ABCDEFGHIJKLMNOPQRSTUVWYZX ABCDEFGHIJKLMNOPQRSTUVWZXY ABCDEFGHIJKLMNOPQRSTUVWZYX ABCDEFGHIJKLMNOPQRSTUVXWYZ ABCDEFGHIJKLMNOPQRSTUVXWZY ABCDEFGHIJKLMNOPQRSTUVXYWZ ABCDEFGHIJKLMNOPQRSTUVXYZW . . . ZYXWVUTSRQPONMLKJIHGFECBAD ZYXWVUTSRQPONMLKJIHGFECBDA ZYXWVUTSRQPONMLKJIHGFECDAB ZYXWVUTSRQPONMLKJIHGFECDBA ZYXWVUTSRQPONMLKJIHGFEDABC ZYXWVUTSRQPONMLKJIHGFEDACB ZYXWVUTSRQPONMLKJIHGFEDBAC ZYXWVUTSRQPONMLKJIHGFEDBCA ZYXWVUTSRQPONMLKJIHGFEDCAB ZYXWVUTSRQPONMLKJIHGFEDCBA We have three questions: 1. What permutation immediately precedes ABCDEFGHIJKLMNOPQZYXWVUTSR? 2. What permutation immediately follows ABCDEFGHIJKLMNOPQZYXWVUTSR? 3. How far apart are AZBCDEFGHIJKLMNOPQRSTUVWXY and ZABCDEFGHIJKLMNOPQRSTUVWXY in the list?

How many sequences of consecutive positive integers add to 1,000,000? List them.

A man is running on a railroad trestle bridge. He has traversed .6 of the length of the bridge when he spots a train ahead of him coming at him. If he continues to run toward the train at a constant speed of 15 miles per hour, the train and he will meet at the end of the bridge. If, however, he instantaneously turns around and goes back from whence he came at a constant speed of 15 miles per hour, the train and he will meet at the other end of the bridge. Assuming that the train's speed is constant, what is the speed of the train?

Find a simple closed form expression for the alternating series of squares, n2-(n-1)2+(n-2)2-(n-3)2+...+42-32+22-12.

Funny, but you can't even do it with a 4 by 4 grid using the numbers 1-16, or even 0-15!

Thanks, Plasmid. I haven't had time to go through this thoroughly, but it looks pretty good so far. You've solved one of the little mysteries which have been nagging me for years!

Thanks for the compliment, CaptainEd! Making a variant of my puzzle says, to me, that you liked mine. This makes me happy indeed. I have a question about going through a line. Let's take the line from (18,19)-(18,100), for example. Suppose, I'm at (16,20) and I'm using a velocity vector of (3,-3). I will end up at (19,17). There can be two interpretations: 1. If I draw a line between the points (16,20) and (19,17), it will not intersect (18,19)-(18,100). 2. I could argue, I moved in the X-direction first, going from (16,20) to (17,20) to (18,20) to (19,20) [thereby crossing (18,19)-(18,100)], THEN moved in the Y-direction from (19,20) to (19,19) to (19,18) to (19,17). Which interpretation of crossing is valid?

Let's have a race in the cartesian plane. We define a point as a pair, (x,y), where x and y are both integers. We will start the race at the point (0,0). Then, we stop at the points (18,20), (14,12), (35,25), (30,8), then back to (0,0), in that order. In order to tell how we can move each time, we have a movement vector (a,b). if we are at point (x,y) and the movement vector is (a,b), the next point we will visit is (x+a,y+b). The movement vector starts at (0,0). Before each move, we may add or subtract 1 from either or both or none of the elements of the movement vector, then we use it to move. To move from (0,0) to (18,20), for example, we can move as follows: 01. at point (0,0); movement vector=(0,0); new movement vector=(-1,1); new point=(-1,1) 02. at point (-1,1); movement vector=(-1,1); new movement vector=(-2,2); new point=(-3,3) 03. at point (-3,3); movement vector=(-2,2); new movement vector=(-3,2); new point=(-6,5) 04. at point (-6,5); movement vector=(-3,2); new movement vector=(-2,3); new point=(-8,8) 05. at point (-8,8); movement vector=(-2,3); new movement vector=(-1,4); new point=(-9,12) 06. at point (-9,12); movement vector=(-1,4); new movement vector=(0,3); new point=(-9,15) 07. at point (-9,15); movement vector=(0,3); new movement vector=(1,3); new point=(-8,18) 08. at point (-8,18); movement vector=(1,3); new movement vector=(2,2); new point=(-6,20) 09. at point (-6,20); movement vector=(2,2); new movement vector=(3,1); new point=(-3,21) 10. at point (-3,21); movement vector=(3,1); new movement vector=(4,1); new point=(1,22) 11. at point (1,22); movement vector=(4,1); new movement vector=(5,0); new point=(6,22) 12. at point (6,22); movement vector=(5,0); new movement vector=(6,-1); new point=(12,21) 13. at point (12,21); movement vector=(6,-1); new movement vector=(6,-1); new point=(18,20) In this example, it took 13 moves to get from (0,0) to (18,20). Our next target is (14,12), but we are stuck with our movement vector of (6,-1) which we can change before we use it to get to our next point. The problem is to go from (0,0) back to (0,0) stopping at the points (18,20), (14,12), (35,25), and (30,8) (in that order, along the way) in the fewest number of moves. Are you up to the challenge?

Hi, pengwen! Welcome. Please explain the individual steps leading to your answer. The fun of this site lay in how people get to their solutions.

4 liters of wine is removed from a wine cask and replaced with 4 liters of water. The wine in the cask is thus diluted. This procedure of removing 4 liters from the cask and replacing it with 4 liters of water is done two more times. The diluted wine left in the cask is now .512 the strength of what it was initially. How much wine did the cask originally hold?

Thanks for all the wonderful work you all have done on this problem. I think we all learned a lot. Now, as promised a few days ago........ Here's what someone told me many years ago on how to uniformly generate a discrete distribution of dimension N: Step 1: Generate N numbers, Xt, t=1,2,3,...,N, chosen randomly from the uniform distribution on (0,1]. Step 2: For t=1,2,3,...,N form Yt = -log(Xt). Step 3: Let S be the sum of all the Yt, t=1,2,3,....,N Step 4: For t=1,2,3,...,N form Zt = Yt÷S The claim is that the resulting distribution, (Z1,Z2,Z3,...,ZN), is what is desired. My simulations make it look true. Can anyone prove or disprove this method?

I would like some clarification. When you say 'generalize to higher dimensional space', do you mean 1) Given a random number generator that uniformly produce numbers between 0 and 1 and a description of ANY high dimensional region (e.g., a sphere in 3D, a circle in 2D, a triangle in a plane embedded within 3D space as described in the OP, etc.), produce a method that uniformly sample coordinates within the described high dimensional region. or 2) Given a random number generator that uniformly produce numbers between 0 and 1, uniformly sample the vector (x1, ...,xn ), where x1 + x2 + ... + xn = 1 and 0< xi < 1 for i = 1, 2, ..., n? I mean #2. I'm sorry that I wasn't clear about that.

I put this problem up because someone told me a simple method of uniformly generating discrete distributions of any dimension. At the time, I was concerned so much about filling my immediate need for such a thing, that I didn't bother asking for a proof of the method. I had hoped that someone on the Den might know of it. I was very happy that you guys did such a nice job analyzing the problem and coming up with the solutions that you did. But, so far, nobody has come up with the method I was given years back. If nobody comes up with it within a few days, I'll post it and ask if anyone can prove/disprove that it works. Thanks!

But I still have not had the patience to write an Excel sheet that tests whether that generates a uniform three dimensional distribution. Edit: fixed spoiler... er, well, I tried to @plasmid: Can you generalize this to higher dimensions?

It can be done without the information to which you refer. Remember, the problem is not to optimize return -- it only asks if either player can guarantee getting at least half the value of the coins.

Sorry, I wanted the picture INSIDE the spoiler, but don't see how to do that.