Added February 2021

I've christened this type of maze a **ring maze** because the moves take place within an algebraic structure that mathematicians call a ring. Don't let this fancy terminology put you off though because it is not necessary to have a particular understanding of mathematics to solve this puzzle. If I haven't scared you away already, then you should find this maze an enjoyable challenge.

## Maze rules:

- Get from the
**0**at the top-left to the**5**at the bottom-right, following a valid sequence of numbers. - Moves can be made
**horizontally**or**vertically**along any rank or file. - A move is valid if you can form a particular equation involving the number you've just left, the number you are moving to and
**one**of the numbers you've crossed in the process of making the move. This is easier than it sounds and I'll give plenty of examples below. **Blank spaces**can be completely**ignored**. Moves can pass over them as if they weren't there and it is not possible to end your move on a blank space.- For clarity, I'll denote the number you've just left as
**x**; the number you've crossed on the way as**y**(there may be more than one option); and the number that you moved to as**z**. - On the first move, you are looking for an equation of the form
**x + y = z**. For example, on the first move you could move across the top row to the 9^{th}and final column by forming the equation**0 + 3 = 3**. Here, you could use any of the intervening**3**s to represent**y**– it doesn't matter. - On the second move, you are looking for an equation of the form
**x × y = z**. - On the third move it is
**x − y = z**. - On the fourth move it is
**x ÷ y = z**. - Then the same +, ×, −, ÷ pattern repeats again
*ad infinitum*, and we go back to**x + y = z**for the fifth move. - The large purple circle to the right of the puzzle helps you keep track of which of the four mathematical operations (+, ×, −, ÷) is relevant for the given move.
- If you add, subtract or multiply two numbers together and end up with a result that is
**not**in the range**0-9**, then you need to throw away any**tens**, and keep only the**units**. For instance**8 + 9 = 17**in standard algebra, but here we throw away the ten and get**8 + 9 = 7**. Similarly,**3 × 6 = 8**because we've thrown**10**away from the usual answer of**18**. - What about, e.g.,
**3 - 5**? This is usually**-2**, but here we just write it as**8 - 10**, throw away the**10**and end up with**8**as the answer – simple. Put even more simply: if you end up with a negative number, then just add**10**to it! - Finally, what do we do about division? We define it by simply multiplying both sides of
**x ÷ y = z**by**y**to yield**x = y × z**instead. If you have no interest in maths, then you can simply use this last equation and completely forget about any complicated notions of division. For example,**8 × 9 = 2**in our alegebra, so we can write**2 ÷ 9 = 8**. - One note of warning: division defined in this way is not uniquely defined and the expression
**x ÷ y**may have more than one answer! If I were a mean sort of a person, then I might use this property to try to lead you astray at some point in the maze. - Finally, as an optional technical aside for maths geeks, this algebraic structure is classed as a
**ring**in mathematics, rather than a**field**, precisely because its concept of division is not uniquely defined. More formally, this algebra is the*ring of integers modulo 10*(**Z/10Z**for short). An interesting fact is that the ring of integers modulo n (**Z/nZ**) is a**field**– with a uniquely defined notion of division – if and only if n is a prime number. Anyway, I think that's enough maths now...

If maths isn't your forte, then you can turn on **lazy mode** by using the checkbox to the right of the maze grid. This will highlight all the possible legal moves in yellow, allowing you to make faster progress and forget about the maths. Unfortunately, I've found that using this mode encourages speculative clicking, which can ultimately make the maze **harder**, rather than easier, to solve. In my opinion, having to concentrate fully on each move is more efficient in the long term because it allows you to develop a more detailed mental map of the paths running through the maze.

Having trouble? You can view the if you are really struggling. This will ruin the puzzle for you, so I would strongly advise against giving up too early!