Varsity Math, Week 110

Varsity Math 110

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The team enjoys playing with these digit combination challenges posed by the coach.

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Math Symbol Balls

Making 25

Using only +, -, ×, ÷, exponents, decimal points, parentheses, and concatenation (that is, combining two digits into another number; for instance, putting 1 and 2 together to make 21 or 12), find two ways to make 25 using 1, 4, and 6. No roots, factorials, repeating decimals, or other math functions are permitted.

Two Ways to 32

Now try making 32 two ways: first using 4, 5, and 7, then using 5, 6, and 7.

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Solutions to week 109

In Commuters’ Dilemma, the equilibrium driving time is 140 minutes. In Construction Conundrum, the equilibrium driving time is 130 minutes.

Commuters’ Dilemma answer explained: There are four paths a commuter can consider: HAS, HABS, HBS and HBAS. The HBAS path takes a constant time of 160 min and will never be chosen because it will always take longer than the others. Suppose J cars take path HAS, K cars take path HABS and L cars take path HBS, where J+K+L = 19,000. Then the commute times for these three paths are:

tJ = (J+K)/200 +50
tK = (J+K)/200+10+(K+L)/100
tL = 100+(K+L)/100

In equilibrium, these three times will be the same, giving J = 15,000, K = 3,000 and L = 1,000
so that tJ = tK = tL = 140 min.

Construction Conundrum answer explained: During the week of construction suppose J cars take path HAS, K cars take path HABS and L cars take path HBS, where J+K+L = 19,000. Then the commute times for these three paths are:

tJ = (J+K)/200 +50
tK = (J+K)/200+20+(K+L)/100
tL = 100+(K+L)/100

In equilibrium, these three times will be the same, giving J = 16,000, K = 0 and L = 3,000
so that tJ = tK = tL = 130 min.

It’s a remarkable fact that the 10-minute extra delay on AB owing to construction decreases everyone’s commute time by 10 minutes. This is an example of what is known as Braess’s Paradox.

Recent Weeks

Week 109: Commuters’ Dilemma & Construction Conundrum, solutions to Nine Coins & Five Questionable Coins

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

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

Come back next week for answers and more puzzles.