# Today Puzzle #580 ## Puzzle No. 580 – Wednesday 2 October

On the M60 at Manchester, I pass 12 junctions when travelling directly between junctions 17 and 4. Between junctions 17 and 3, I also pass 12 junctions. How many junctions are on the M60?

Today's #PuzzleForToday has been set by David Baynard, a biotechnology PhD student at Emmanuel College, Cambridge

To calculate the number of junctions passed between the two given sequentially numbered junctions, subtract 1 from their difference. The difference gives the number of stretches between junction numbers used in the calculation; this is one more than the number of junctions passed.

17 - 4 - 1 = 12

17 - 3 - 1 = 13 ≠ 12

Wait, what?

The clue is the question: how many junctions are on the M60? The only way the question has enough information to solve the problem is if the M60 is a ring road, and so the direct route between junctions 17 and 3 passes junction 1.

This means that the difference is 3 - 17, rather than 17 - 3.

3 - 17 - 1 = -15 ≠ 12

If there are n junctions, then these differences should be calculated using modular arithmetic based on modulus n. That means if the calculation results in a number less than 0, or greater than or equal to n, add or subtract n until it falls within the range. This leads to the following equation:

-15 mod n = 12

which can be solved by trying

-15 + n = 12

And so the number of junctions, n = 27  