We’re thrilled that Ritangle 2022, which ran between October and December last year, was our biggest since the competition began seven years ago! More than 1000 teams of young mathematicians across the world put their problem-solving skills to the test in a bid to become Ritangle 2022 champions!

Over the course of nine weeks, participants had the opportunity to solve 28 challenges. Every correct answer revealed a piece of information that helped to crack a code and unlock the final task; 292 teams managed this – an amazing achievement!

On 6 December, the race was then on to be the first team to submit the best answer to the final question. We received answers from 145 teams and, after very careful checking from our judges, we were delighted to announce that the winners of our Ritangle 2022 competition are from The Liverpool Blue Coat School.

Congratulations to the winning team, we hope you enjoy your fantastic Ritangle prize hamper which includes a free one-year subscription to Integral!

We’ve received some great feedback this year, including this message from a team who created a brilliant interactive tool to help them solve the final task:

Dear Ritangle Team,

Thanks for hosting such a great competition this year!

Our team really enjoyed finding a solution for the Stage 3 question and although we didn’t quite get the fastest solution, we did have fun building tools to help us. We thought these might be beneficial for people looking back through previous Stage 3 questions or just for your team if you are still looking for a faster solution (if there is one!).

We started out with generating a list of mountains and their coordinates using a simple Python script, before plotting them on a set of axes Desmos. This was useful to visualise the distribution of them but not for calculating the time taken when going from one peak to another.

After solving the trigonometric equation to find the gradient (0.75) and therefore the speed along the flat, uphill and downhill sections, we found formulas for the time taken to travel between 2 peaks. We also found algorithms to check if any given transition was allowed (not spawning inside mountains). This meant we had enough to write a program to calculate the time taken for any given order of peaks, but it would still require us manually entering coordinates and was hard to visualise. We realised very early on that finding the shortest solution by brute forcing 60! permutations was not going to work as we didn’t have infinite computing power, so the solution had to be found by human intuition (at least in part).

Enter our interactive, online solution finder. The main attraction is a 28×28 grid representing the island, with mountain peaks represented by dots, all inside an outline dictated by the set of points either on or just outside the upper-right quarter of a 28km radius circle (the coastline). Each peak has a circle around it representing the base of that mountain. The user can click the peaks in sequence to build up a solution. At every step you will only be able to select valid moves (invalid moves will be greyed out) so any path involving all 60 peaks is valid in the eyes of the question.

There are various numerical outputs at the top of the map as well:

 – the top left corner shows the optimum start and end points along the coastline (red represents the first peak visited and green represents the last)

 – the upper number in the middle shows the number of seconds the route would take (including travel to and from the coastline at the start and end), rounded to 3 decimal places

 – below this is the same time but in a more user-friendly format

 – the counter in the top-right shows how many peaks are on the path you are plotting, going from 0 when you have selected no peaks to 60 when you have a valid path

There is also a download button at the foot of the page which downloads a text file with the x and y coordinates of each point along the path (including the start and end points on the coast).

This tool can be viewed at https://ritangle-final.vercel.app/ and will remain live for the foreseeable future. Note that on some devices, especially ones with smaller screens, the browser might need to be zoomed in or out for the map to become interactive. It’s best viewed on a desktop computer with a large screen.

Ben Sommer

If you’d like to view the questions from this year’s competition, please visit our Ritangle 2022 page.