You just made a new friend at an international puzzle conference, and you asked for a way to keep in touch. You found the following note slipped under your hotel room door the next day:
"Salutations, new friend! I have replaced every digit of my phone number with its spelled-out uppercase English representation ("ZERO", "ONE", "TWO", "THREE", "FOUR", "FIVE", "SIX", "SEVEN", "EIGHT", "NINE" for the digits 0 through 9, in that order), and then reordered all of those letters in some way to produce a string S. It's up to you to use S to figure out how many digits are in my phone number and what those digits are, but I will tell you that my phone number consists of those digits in nondecreasing order. Give me a call... if you can!"
You are a teacher at the brand new Little Coders kindergarten. You have N kids in your class, and each one has a different student ID number from 1 through N. Every kid in your class has a single best friend forever (BFF), and you know who that BFF is for each kid. BFFs are not necessarily reciprocal -- that is, B being A's BFF does not imply that A is B's BFF.
Your lesson plan for tomorrow includes an activity in which the participants must sit in a circle. You want to make the activity as successful as possible by building the largest possible circle of kids such that each kid in the circle is sitting directly next to their BFF, either to the left or to the right. Any kids not in the circle will watch the activity without participating.
What is the greatest number of kids that can be in the circle?
When Sergeant Argus's army assembles for drilling, they stand in the shape of an N by N square grid, with exactly one soldier in each cell. Each soldier has a certain height.
Argus believes that it is important to keep an eye on all of his soldiers at all times. Since he likes to look at the grid from the upper left, he requires that:
Within every row of the grid, the soldiers' heights must be in strictly increasing order, from left to right.
Within every column of the grid, the soldiers' heights must be in strictly increasing order, from top to bottom.
Although no two soldiers in the same row or column may have the same height, it is possible for multiple soldiers in the grid to have the same height.
Since soldiers sometimes train separately with their row or their column, Argus has asked you to make a report consisting of 2*N lists of the soldiers' heights: one representing each row (in left-to-right order) and column (in top-to-bottom order). As you surveyed the soldiers, you only had small pieces of paper to write on, so you wrote each list on a separate piece of paper. However, on your way back to your office, you were startled by a loud bugle blast and you dropped all of the pieces of paper, and the wind blew one away before you could recover it! The other pieces of paper are now in no particular order, and you can't even remember which lists represent rows and which represent columns, since you didn't write that down.
You know that Argus will make you do hundreds of push-ups if you give him an incomplete report. Can you figure out what the missing list is?
On the game show The Last Word, the host begins a round by showing the contestant a string S of uppercase English letters. The contestant has a whiteboard which is initially blank. The host will then present the contestant with the letters of S, one by one, in the order in which they appear in S. When the host presents the first letter, the contestant writes it on the whiteboard; this counts as the first word in the game (even though it is only one letter long). After that, each time the host presents a letter, the contestant must write it at the beginning or the end of the word on the whiteboard before the host moves on to the next letter (or to the end of the game, if there are no more letters).
For example, for S = CAB, after writing the word Con the whiteboard, the contestant could make one of the following four sets of choices:
put the A before C to form AC, then put the B before AC to form BAC
put the A before C to form AC, then put the B after AC to form ACB
put the A after C to form CA, then put the B before CA to form BCA
put the A after C to form CA, then put the B after CA to form CAB