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Four glasses on a square table

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four glasses on a square table puzzle, also known as the blind bartender’s problem

Four glasses are placed on the corners of a square table. Some of the glasses are upright (up) and some upside-down (down). You have to arrange the glasses so that they are all up or all down (while keeping your eyes closed all the time). The glasses may be re-arranged in turns subject to the following rules.

  1. Any two glasses may be inspected in one turn and after feeling their orientation you may reverse the orientation of either, neither or both glasses.
  2. After each turn table is rotated through a random angle.
  3. At any point of time if all four glasses are of the same orientation a ring will bell

You have to come up with a solution to ensure that all glasses have the same orientation (either up or down) in a finite number of turns. The algorithm must be non-stochastic i.e. it must not depend on luck.
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Fair and Square Solution: Google codejam 2013 Qual Round

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Most of us are familiar with google codejam, those who don’t visit https://code.google.com/codejam for more information. This is the third problem from Qualification Round 2013.

Problem Statement

Little John likes palindromes, and thinks them to be fair (which is a fancy word for nice). Apalindrome is just an integer that reads the same backwards and forwards – so 6, 11 and 121 are all palindromes, while 10, 12, 223 and 2244 are not (even though 010=10, we don’t consider leading zeroes when determining whether a number is a palindrome).

He recently became interested in squares as well, and formed the definition of a fair and square number – it is a number that is a palindrome and the square of a palindrome at the same time. For instance, 1, 9 and 121 are fair and square (being palindromes and squares, respectively, of 1, 3 and 11), while 16, 22 and 676 are not fair and square: 16 is not a palindrome, 22 is not a square, and while 676 is a palindrome and a square number, it is the square of 26, which is not a palindrome.

Now he wants to search for bigger fair and square numbers. Your task is, given an interval Little John is searching through, to tell him how many fair and square numbers are there in the interval, so he knows when he has found them all.
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Lawnmower Solution: Google codejam 2013 Qual Round

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Most of us are familiar with google codejam, those who don’t visit https://code.google.com/codejam for more information. This is the second problem from Qualification Round 2013.

Problem

Alice and Bob have a lawn in front of their house, shaped like an N metre by M metre rectangle. Each year, they try to cut the lawn in some interesting pattern. They used to do their cutting with shears, which was very time-consuming; but now they have a new automatic lawnmower with multiple settings, and they want to try it out.

The new lawnmower has a height setting – you can set it to any height h between 1 and 100 millimetres, and it will cut all the grass higher than h it encounters to height h. You run it by entering the lawn at any part of the edge of the lawn; then the lawnmower goes in a straight line, perpendicular to the edge of the lawn it entered, cutting grass in a swath 1m wide, until it exits the lawn on the other side. The lawnmower’s height can be set only when it is not on the lawn.

Alice and Bob have a number of various patterns of grass that they could have on their lawn. For each of those, they want to know whether it’s possible to cut the grass into this pattern with their new lawnmower. Each pattern is described by specifying the height of the grass on each 1m x 1m square of the lawn.

The grass is initially 100mm high on the whole lawn.
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Tic-Tac-Toe-Tomek Solution: Google codejam 2013 Qual Round

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Most of us are familiar with google codejam, those who don’t visit https://code.google.com/codejam for more information. This is the first problem from Qualification Round 2013.

Problem Statement

Tic-Tac-Toe-Tomek is a game played on a 4 x 4 square board. The board starts empty, except that a single ‘T’ symbol may appear in one of the 16 squares. There are two players: X and O. They take turns to make moves, with X starting. In each move a player puts her symbol in one of the empty squares. Player X’s symbol is ‘X’, and player O’s symbol is ‘O’.

After a player’s move, if there is a row, column or a diagonal containing 4 of that player’s symbols, or containing 3 of her symbols and the ‘T’ symbol, she wins and the game ends. Otherwise the game continues with the other player’s move. If all of the fields are filled with symbols and nobody won, the game ends in a draw. See the sample input for examples of various winning positions.

Given a 4 x 4 board description containing ‘X’, ‘O’, ‘T’ and ‘.’ characters (where ‘.’ represents an empty square), describing the current state of a game, determine the status of the Tic-Tac-Toe-Tomek game going on. The statuses to choose from are:

  • “X won” (the game is over, and X won)
  • “O won” (the game is over, and O won)
  • “Draw” (the game is over, and it ended in a draw)
  • “Game has not completed” (the game is not over yet)

If there are empty cells, and the game is not over, you should output “Game has not completed”, even if the outcome of the game is inevitable.
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