Each year Integral Maths runs the Ritangle competition, and it is by far my favourite maths competition. UKMT Maths Challenges cater for those who love time-pressured competitions, letting learners pit their wits, intuition and mathematical flexibility against one another. Their Team Challenge incorporates collaboration, and is hugely enjoyable, but if anything even more time-pressured and those who excel at rapid mental or written calculation undoubtedly have an edge. Ritangle, on the other hand, caters to a much broader cohort of budding mathematicians: the fast and the slow, the intuitive and the methodical, the pen-and-paper wizard and the technology geek.

Ritangle encourages the kind of approach I like to take when solving an interesting mathematical problem: I don’t set myself time limits – I often niggle away at a knotty problem for weeks; I don’t work alone – if I get stuck, I seek inspiration or advice, or a new perspective from someone else; I don’t needlessly ignore the benefits of technology or online research.

This post is just a snapshot of a few of my favourite Ritangle questions from previous years, saying what I liked about them and how I solved them. I’ve deliberately picked out those questions which lend themselves well to the ‘use of technology’, which is another reason I like the competition so much: students who have never done any coding, or even used a spreadsheet, will see the usefulness of these tools for solving certain types of problem, and the emphasis I place on verifying results independently is hopefully one which they carry back into their A-level Maths lessons and beyond. Instead of just answering a question one way, then asking the teacher if it’s right, find ways to check your result yourself – there are often quick things you can check, but given the time, there’s no substitute for tackling the problem in two entirely different ways and comparing your results. I will link to GeoGebra files, Google Spreadsheets or Python code I have used. If you solved one of these problems using a different approach, or if there’s another ‘technology-friendly’ Ritangle question you’d like me to share my method for, leave a comment. Just remember that discussing any current Ritangle competition questions goes against the spirit of the competition, so if your comment mentions any it won’t get published (at least, not for a few more months!)

Without further ado, the first question, from 2017:

**The Ants Go Marching (2017 Q4)**

**Why I like it**

Ants and Geometric Series: what’s not to love? One thing I like about it is the visual nature of the problem. Solvers can quickly get a sense of what is happening, and how an ant following an infinite set of instructions could ever ‘end up’ anywhere, by simply plotting the first few points (either by hand or with GeoGebra). I also like the fact that solvers can break down the moves in a number of different ways: the north-south moves of each ant form their own geometric progression, independent of the east-west moves, for instance. But if we can treat the north-south moves as one single GS (just with a negative common ratio), what’s to stop us treating *all *moves as a single GS (with an *imaginary *common ratio)?

**How to solve it**

We can break apart each ant’s moves into North-South and East-West. By using the infinite sum formula for a geometric series, we can find the eventual position of each ant. Alternatively, notice that each move can be described by a complex number: the first ant’s moves can be treated as a geometric series with common ratio 0.9i, and (although the numbers aren’t much more elegant to deal with) it feels somewhat satisfying to deal with an entire two-dimensional journey with a single mathematical trick.

**How to check it**

The most obvious way is brute force: try moving the ants for a while and see where they end up. I’ve made a deliberately sub-optimal bit of code to represent this: I’m intentionally making it as close to the original problem as possible, not taking advantage of any mathematical insights such as grouping vertical and horizontal motion separately. This will ensure that if I have made any errors in my assumptions, they won’t pollute this verification process also: Ritangle2017_Q4 Python Snippet.

**Moving Squares (2017 Q6)**

**Why I like it**

It’s a superb example of a problem that looks tricky, but just below the surface is simply a quadratic expression wrapped up in some algebraic fractions. No knowledge beyond the first few weeks of year 12 is required, and students will have the satisfaction of finding a non-obvious solution to an unfamiliar problem without having to lean on any mathematical tools they don’t already have firmly in their grasp.

**How to solve it**

Write each of the four areas in terms of the variable *c*. When you substitute them all into the fraction, the whole thing simplifies, with a bit of work, to the point where all that remains is some careful thinking and completing the square. No calculus required.

**How to check it**

A question like this is crying out for some trial-and-error, preferably via a rough and ready GeoGebra manipulative. GeoGebra handily allows us to not only create movable components and measure them, but also embed calculations, so we can watch as the fraction we want to minimise is evaluated before our eyes, and see if the numbers we found really work.

**Shifting Squares (2017 Q9)**

**Why I like it**

It’s a nice question for beginner coders to cut their teeth on. The numbers are clearly too big to yield their secrets to a lucky guess, or even to a concerted effort with a pocket calculator. We’ll need variables, conditionals and loops, but this is an ideal example of a question where coding really helps: there’s no fancy method, just do something basic over and over again until we get the result required.

**How to solve it**

We’ll need to define our variables, then loop through a few thousand possible values for d, and test the various sums to see if they’re all square. If they are, give the result: Ritangle2017_Q9 Python Snippet

**How to check** **it**

Just like coding trivialises the process of finding solutions, the nature of the problem means that verifying the solution is also pretty straightforward: bung the numbers into your calculator and see if it works.

**Digitally Challenged (2018 Preliminary Question 1)**

**Why I like it**

There are lots of ways to solve this. We can make some headway almost immediately simply by trying a few likely numbers. The more examples we find, the more patterns we are likely to spot, and it’s a question that lends itself well to different members of the team playing to their various strengths: someone can code it up in Python, while someone else uses Excel, and another person plays around with laws of divisibility and combinations. Each method has its own pros and cons, and the question is a good way to explore them, and to find ways for the different approaches to complement one another.

**How to solve it**

Brute force, with Python: looping through all 8 digit numbers takes a while. Checking to see what digits a number is made from is sufficiently complex for a computer (converting from a binary form to decimal, then extracting digits either by modular arithmetic or converting to text strings), that it pays to optimise your code somewhat. Only looping through multiples of 18 helps. Restricting your range to values starting at 11111112 and ending with 33333333 is another easy win. Will anyone spot that we can loop through multiples of 90 without losing any relevant numbers? Ritangle2018_P1 Python Snippet.

Base 3, with Excel: there are nearly 100 million 8-digit numbers in base 10, but only 6,561 8-digit combinations of three digits. A nice lazy way to create them is to make use of the BASE function, which can convert a decimal like 6561 into a base-3 number like 22222222. We can then swap out all the zeroes for 3s, and treat them as decimal numbers. Test for division using the MOD function, and Bob’s your uncle. Ritangle2018_P1 Spreadsheet.

Careful counting, by hand: If a number divides by 18, it must also divide by 2. If the only allowed digits are 1, 2 or 3, it must end in a 2. Also, since it divides by 18, it divides by 9, and therefore the digits add up to a multiple of 9. Since we know already it ends in a 2, that means the first 7 digits combined have to add up to either 7, 16 or 25, etc. But since they can’t be any greater than 3s, the most they can add up to is 21, which means there are only two cases to consider. The first case is simple: 11111112. The second takes some more work, but will yield nicely to a systematic approach. Notice that the order of the digits won’t matter: if the digit sum is right, it will divide by 9. Try finding all the ways to have the seven digits add to 21 with as many 3s as possible, then gradually reduce them. The binomial *choose *or *combinations *function may come in handy. For example, four 3s, one 2 and two 1s is a valid combination. There are *7 choose 4 *ways to place the 3s, followed by 3 ways to place the 2, then the 1s must go in the remaining two places.

**How to check** **it**

My approach was to use all three methods described above. When they all yield the same answer, that’s pretty convincing. It’s also worth noting that, even partial success with, say, Python or Excel, can give fresh insights (“Why do they all end in a 2?” etc), and if you look closely at the results the Python code spits out, you can see the various rearrangements of the digits just as the by-hand method predicts.

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