Problem 7 · 2012 Math Kangaroo
Easy
Logic & Word Problems
path-tracing
Each of the nine paths in a park is 100 m long. Anna wants to walk from A to B without using the same path twice. How long is the longest path she can choose?
Show answer
Answer: C — 700 m
Show hints
Hint 1 of 3
Try to walk along as many paths as you can without repeating one.
Still stuck? Show hint 2 →
Hint 2 of 3
Count how many paths meet at each junction — a route can pass straight through a junction only if an even number of paths meet there.
Still stuck? Show hint 3 →
Hint 3 of 3
Anna's start A and finish B are different corners, so she can't end where she began — check whether using all paths is even possible.
Show solution
Approach: count odd junctions (Euler trail)
- Each junction needs paths to enter and leave in pairs; only the start and the finish are allowed to have an odd number of paths meeting them.
- In this triforce network every one of the six junctions has an even number of paths (the three corners have 2 each, the three midpoints have 4 each), so a single route cannot start at corner A and end at the different corner B while using all 9 paths.
- Leaving out the side that joins A to B (its two 100 m pieces) makes A and B the only odd junctions, and then all 7 remaining paths can be walked in one go.
- That gives 7 × 100 = 700 m, answer C.
Mark:
· log in to save