Varsity Math 109


The coach can take several different routes to school each morning, and challenges the team to find the most efficient path.



Commuters’ Dilemma

The road network shown below accommodates 19,000 cars during eastbound rush-hour traffic each morning, with each driver trying to get from the area where the coach lives (home, or H), to the vicinity of the school (S) in the least amount of time. Any individual driver will change routes if the new route reduces total travel time. The driving time for each leg is shown in minutes. Note that the driving times for legs HA and BS depend on the number of cars, N (out of 19,000), electing to take that leg.

When the system is in equilibrium, what is the commuting time from H to S?

Construction Conundrum

Suppose that during one week, because of construction on the connector road from A to B, there is a predictable delay so that the time on that leg becomes 20 minutes (instead of 10).

Now what does the equilibrium driving time become for the commute from S to F?


Solutions to week 108

In Nine Coins, two weighings get the job done. In Five Questionable Coins, Ethan can label the coins A to F, with F being the true coin. First weigh B+F against C+E; then D+F against B+E.

Nine Coins answer explained: Two weighings get the job done. One way to do this is to weigh 33 vs 2111, leaving out coins 322. This establishes the first two as 3 and 3. Take one of them with the “left out” 3 and weigh it against a “left out” 2 and 111. This establishes all the 3’s and all the 1’s. The 2’s are established by remainder.

Five Questionable Coins answer explained: One way to do this is to label the coins from A to F, with F being the true coin. In the first weighing, balance B+F (right pan) against C+E (left pan). In the second weighing, balance D+F (right pan) against B+E (left pan). If both weighings balance, then coin A is fake. If the two weighings are unbalanced in opposite directions, then coin B is fake. If the two weighings are unbalanced in the same direction, then coin E is fake. If the first weighing balances and the second doesn’t, then coin D is fake. If the first weighing doesn’t balance and the second does, then coin C is fake.

Recent Weeks

Week 108: Nine Coins & Five Questionable Coins, solutions to Ten from Two & Triangular Boundary

Week 107: Ten from Two & Triangular Boundary, solutions to Simple Multiple & Small and Simple

Week 106: Simple Multiple & Small and Simple, solutions to Math for the Ages & Three Racers

Week 105: Math for the Ages & Three Racers, solutions to X Factor & Perfect Pairings

Week 104: X Factor & Perfect Pairings, solutions to Two Sums & Double-Faced

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

Come back next week for answers and more puzzles.