Schools Mathematics Grand Challenge
PROBLEM 19:
One of the most famous sequences of numbers in mathematics is the
"Fibonacci sequence". The rules that describe this famous sequence of
numbers are the following:
 The first number in the sequence is 1;
 The second number in the sequence is 1;
 Every number from the third number on is the sum of the previous
two numbers  so the third number is 1 + 1 = 2, the fourth number is 2 +
1 = 3 and so on.
So, the first few numbers in the Fibonacci sequence are:
1, 1, 2, 3, 5, 8, 13, 21
What is the units digit of the 1200th number in the Fibonacci sequence?
PROBLEM 20:
The queen domination problem (QDP for short) is a problem in mathematics
inspired by chess. On a normal chessboard with 64 squares the QDP may
be phrased as follows: given a certain number of queens, place them on
the board in such a way that each of the 64 squares is either covered
by a queen or under attack by a queen.
We say that a square is under attack (by a queen) if it is contained in
the set of possible queen moves. Recall that a queen can move any number
of squares in a straight line vertically, horizontally, or diagonally,
as illustrated below. Thus, in this example, the squares under attack
are the squares marked with solid yellow circles. The starshaped symbol
respresent a square occupied by a queen.
Going back to our queen domination problem now. The picture below shows
a solution to a QDP for a chessboard with 5x5 squares. For this case,
the number of queens, 3, is the smallest number of queens that yields
a solution to the QDP. In other words: there is no way of 'dominating'
a 5x5 chessboard with 2 queens (or 1 for that matter).
Now consider the`chessboard' below and answer the following question:
what is the minimum number of queens needed to occupy or attack all 92
squares on this modified chessboard.
HINT: the miminum number of queens needed to solve the
QDP for a regular 8x8 chessboard is 5.
PROBLEM 21:
If n is a whole number, then
M = 4^{40} + 4^{1024} + 4^{n}
is also a whole number. However, M isn't always the square of a whole
number. What is the the biggest n so that M is the square of a
whole number?
NOTE: In this question, 4^{40} means 4 multiplied
by itself 40 times, 4^{1024} means 4 multiplied by itself 1024
times and 4^{n} means 4 multiplied by itself n times.
HINT: if no one finds the biggest possible n, we will
give credit for the largest submitted n that makes M the square
of a whole number.
