superprismatic

I wrote a program which computes all binary matrices for each pair of (rows,columns) in the following table. It then computes the number of these which have ledft-to-right paths of 1s as specified in the OP. I hope this helps those of you who are working on this problem. As for me, I don't see a good way to approach the problem analytically. rows columns # with paths # matrices possible % having a path 2 8 1393 65536 2.126% 3 8 571992 16777216 3.409% 2 7 577 16384 3.522% 3 7 116472 2097152 5.554% 2 6 239 4096 5.835% 3 6 23624 262144 9.012% 2 5 99 1024 9.668% 4 6 2089896 16777216 12.457% 3 5 4768 32768 14.551% 2 4 41 256 16.016% 4 5 204852 1048576 19.536% 3 4 956 4096 23.340% 5 5 8176228 33554432 24.367% 2 3 17 64 26.563% 4 4 19864 65536 30.310% 5 4 385064 1048576 36.723% 3 3 190 512 37.109% 6 4 7141192 16777216 42.565% 2 2 7 16 43.750% 4 3 1896 4096 46.289% 5 3 17744 32768 54.150% 3 2 37 64 57.813% 6 3 159552 262144 60.864% 7 3 1396608 2097152 66.595% 4 2 175 256 68.359% 8 3 11993600 16777216 71.487% 5 2 781 1024 76.270% 6 2 3367 4096 82.202% 7 2 14197 16384 86.652% 8 2 58975 65536 89.989% [/code]

This isn't true. Here's a counterexample: 101 100 011 Here, there is no path of zeros going from top to bottom. But, there is also no path of ones going from left to right.

I made the stupid mistake of having everything in there twice. So, just take the first (or last) half of my answers.

If we were to write the numbers from 1 to 2001 in Roman numeral notation (see http://www.novaroma.org/via_romana/numbers.html ) and sort this list alphabetically, what would be the number in the middle (1001st in the list)? You can use the converter on the web site to check the way you form numbers using Roman numerals.

Having given a shamefully incorrect knee-jerk answer to howardl1963's great little puzzle, "Guessing Coins", I would like to offer a slight modification which, I think, points out the richness of this little gem. I hope howardl1963 doesn't mind that I have taken his statement very nearly verbatim: Bob offers Jim and Jack a gamble. Jim and Jack are allowed to discuss a strategy before hand, but won't be able to communicate in any way afterwards. Jim and Jack will be put in separate rooms, with 1 fair coin each. Bob will go into one room, and ask Jim to flip his coin. He will then ask Jim to predict the result of the upcoming flip by Jack. Bob will then go into Jack's room. He will ask Jack to flip his coin. He will then ask Jack to guess what Jim flipped. If BOTH of them guessed right, they each have to give Bob 10 dollars. If EITHER of them are wrong, Bob will give them 1 dollar each. Should they accept the gamble?

Zathros's 6 day solution in post #7 is one of four optimal solutions. All possible groundhog moves are covered by his solution. Plainglazed got it first but I didn't spot it before editing this post.

It seems to me that 1, 2, 3, and 4 colour the entire plane. There doesn't seem to be anything left over to colour bistre.

I think there's a typo in the problem. The 15712 should be 13712.

The units digit is the least significant digit of the number. In our usual notation, it is the rightmost digit.

If y<1, then there is and it is 1/(1-y).

Smoo Well, one of A or B will win. If, as you say, A's probability of winning is .36, then B's probability of winning is .64 -- larger than that of A, the better player. How strange!

This problem has been posted before. Please read the board guidelines and search before posting.

Ah, But you are neglecting the quantum effects! These are not completely negligible, especially since there must be some differences in the masses of the eggs.