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### 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:

t_{J} = (J+K)/200 +50

t_{K} = (J+K)/200+10+(K+L)/100

t_{L} = 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 t_{J} = t_{K} = t_{L} = 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:

t_{J} = (J+K)/200 +50

t_{K} = (J+K)/200+20+(K+L)/100

t_{L} = 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 t_{J} = t_{K} = t_{L} = 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.