# World Maths Day (3 March) Puzzle

Olivia Miller

Dr Daniel Bearup, Dr Clelia Pech and Professor Peter Hydon from the University’s School of Mathematics, Statistics and Actuarial Science (SMSAS) have developed a problem-solving puzzle for World Maths Day (3 March).

Can you solve ‘The Balanced Table Problem’? The two-part theoretical puzzle is as follows:

A hotel has been booked to host a large international conference. It is expected that the conference delegates will take advantage of meeting to engage in wide ranging bilateral negotiations.

As part of creating a congenial atmosphere for such negotiations the hotel manager has set aside several small rooms which they intend to furnish with a round table and two chairs. To add some aesthetic interest to these otherwise standardised layouts they decide that each table should include a design consisting of a four-by-four grid of tiles in two colours (black and white). However, in order to avoid inducing any subconscious biases, they impose some constraints on the allowable designs.

a) Each row and column of the grids must contain the same number of white and black tiles (i.e. two of each).

b) The two negotiators must each see the same pattern when they are seated at the table (i.e. the right side of the design must be a reflection of the left side).

Challenge 1: How many distinct table designs can be created?

Challenge 2: The hotel manager decides that, given the number of delegates expected, this will not be enough unique designs. To allow a broader range of designs, they introduce an alternative to Constraint b).

a) If the table is rotated 180 degrees, the pattern that the two negotiators see will be unchanged (i.e. the table is rotationally symmetric).

How many distinct table patterns can be created now?

Find out the answers along with a breakdown of the solutions from Dr Bearup here.