Problems

Draw how to tile the whole plane with figures, consisting of squares \(1\times 1\), \(2\times 2\), \(3\times 3\), \(4\times 4\), \(5\times 5\), and \(6\times 6\), where each square appears an equal number of times in the design of the figure. Can you think of two essentially different ways to do this?

Find a non-regular octagon which you can use to tile the whole plane and show how to do that.

Observe that \(14\) isn’t a square number but \(144=12^2\) and \(1444=38^2\) are both square numbers. Let \(k_1^2=\overline{a_n...a_1a_0}\) the decimal representation of a square number.
Is it possible that \(\overline{a_n...a_1a_0a_0}\) and \(\overline{a_n...a_1a_0a_0a_0}\) are also both square numbers?

Multiply an odd number by the two numbers either side of it. Prove that the final product is divisible by \(24\).

Mattia is thinking of a big positive integer. He tells you what this number to the power of \(4\) is. Unfortunately it’s so large that you tune out, and only hear that the final digit is \(4\). How do you know that he’s lying?

You might want to know what day of the week your birthday is this year. Mathematician John Conway invented an algorithm called the ‘Doomsday Rule’ to determine which day of the week a particular date falls on. It works by finding the ‘anchor day’ for the year that you’re working in. For \(2025\), the anchor day is Friday. Certain days in the calendar always fall on the anchor day. Some memorable ones are the following:
‘\(0\)’ of March - which is \(29\)th February in a leap year, and \(28\)th February otherwise.

\(4\)th April, \(6\)th June, \(8\)th August, \(10\)th October and \(12\)th December. These are easier to remember as \(4/4\), \(6/6\), \(8/8\), \(10/10\) and \(12/12\).

\(9\)th May, \(11\)th July, \(5\)th September and \(7\)th November. These are easier to see as \(9/5\), \(11/7\), \(5/9\) and \(7/11\). A mnemonic for them is “9-5 at the 7-11".
Then find the nearest one of these dates to the date that you’re looking for and find remainders.

For example, \(\pi\) day, (\(14\)th March, which is written \(3/14\) in American date notation. It’s also Albert Einstein’s birthday) is exactly \(14\) days after ‘\(0\)’th March, so is the same day of the week - Friday in \(2025\).

What day of the week will \(25\)th December be in \(2025\)?

\(6\) friends get together for a game of three versus three basketball. In how many ways can they be split into two teams? The order of the two teams doesn’t matter, and the order within the teams doesn’t matter.

That is, we count A,B,C vs. D,E,F as the same splitting as F,D,E vs A,C,B.

In the diagram, all the small squares are of the same size. What fraction of the large square is shaded?

The letters \(O\), \(P\), \(S\) and \(T\) represent different digits from \(1\) to \(9\). The same letters correspond to the same digits, while different letters correspond to different digits.

Find \(O+P+S+T\), given that \(SPOT+POTS=15,279\).

Picasso colours every point on the circumference of a circle red or blue. Is he guaranteed to create an equilateral triangle all of whose vertices are the same colour?