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### Schools Mathematics Grand Challenge

PROBLEM 7

Many of you have probably done SUDOKU puzzles before. Consider the SUDOKU puzzle based on the grid shown below. Each horizontal row, each vertical column and each of the four smaller grids (each containing four squares), must contain all of the numbers 1, 2, 3, 4. Usually, there is only one solution to a SUDOKU puzzle. However, in the grid below, there is more than one way of completing the puzzle. How many different completions of the grid are possible?

PROBLEM 8

A large water tank in a country with a perfectly regular climate is exactly 48 feet high. During the rainy season, each day has exactly 12 hours of rainfall followed by 12 hours of evaporation. The water level in the tank increases at a uniform rate of 1 foot per hour during the 12 hours of rainfall, and it decreases at a uniform rate of half a foot per hour during the 12 hours of evaporation. If the tank is empty at the beginning of a 12 hour rainfall period, how many hours will it take for the water level in the tank to reach the top of the tank for the first time?

For example, after 12 hours the water level is 12 feet, while after 24 hours the level is 6 feet.

PROBLEM 9

In English the most common letter is "E". We have made up a message telling you how to answer this problem and in the message the most common letter is E.

However, we have encrypted the message! For every A in the original message, we counted forward a secret number of letters in the alphabet and wrote that down. The same for B, and so on. While we are doing this counting, if we get to Z, the next letter is A.

For example, if the secret number was 2, "zebra" would become "bgdtc". This is called a Caesar Cipher after Julius Caesar who used it to talk to his generals.

In the problem below, the secret number is not 2, it is secret!

Fqjc rb nunenw crvnb bnenwcnnw?