In the much-missed student quiz show Blockbusters, teenagers would ask host Bob Holness for a letter from a hexagonal grid. How we laughed when a contestant asked for a P!
Holness would reply with a question in the following style: What P is…
For example: What P is an area of cutting edge mathematical research and also a process in the making of an espresso?
The answer is the subject of today’s puzzle: percolation.
Percolation is an important research area that emerged from statistical physics and is concerned with how fluids flow through porous materials. French mathematician Hugo Duminil-Copin won the Fields Medal, maths’ most high profile prize, earlier this month for his work in this area.
Today’s perplexing percolation poser concerns the following Blockbusters-style hexagonal grid:
The grid above shows a 10x10 hexagonal tiling of a rhombus (i.e. a diamond shape), plus an outer row that demarcates the boundary of the rhombus. The boundary row on the top right and the bottom left are coloured blue, while the boundary row on the top left and the bottom right are white.
If we colour each hexagon in the rhombus either blue or white, one of two things can happen. Either there is a path of blue hexagons that connects the blue boundaries, such as here:
Or there is no path of blue hexagons that connects the blue boundaries, such as here:
There are 100 hexagons in the rhombus. Since each of these hexagons can be either white or blue, the total number of possible configurations of white and blue hexagons in the rhombus is 2 x 2 x … x 2 one hundred times, or 2100, which is about 1,000,000,000,000,000,000,000,000,000,000.
In how many of these configurations is there a path of blue hexagons that connects the blue boundaries?
The answer requires a simple insight. Indeed, it is the insight on which the quiz show Blockbusters relied.
I’ll be back at 5pm UK with the answer. Meanwhile, NO SPOILERS.
UPDATE: Read the solution here.
The hexagonal grid is a basic model in percolation theory: the path between boundaries represents the ability of a fluid to pass across the grid. I’ll explain more about the relevance and importance of this model in the 5pm post that reveals the answer to the puzzle.
For clarification: a path of hexagons means a sequence of adjacent hexagons that are the same colour.
Thanks to Ariel Yadin, of Ben-Gurion University in Israel, for suggesting this puzzle.
I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.
Here’s a final question: What T is the latest title in the children’s book series Football School that I write with Ben Lyttleton, which is full of hilarious questions about the world’s most popular sport? Yes, it’s the The Greatest Ever Quiz Book, out now!
I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.
