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
Click here for the answer
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
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