## ________________

The coach provides these two reasoning problems to prepare the math team.

## ________________

### Nick’s Birthday

A while back, Nick stated: “Sometime during last year, I was still 21; in two days I’ll be in my 25th year.”

*What day of the year is Nick’s birthday and on what day of the year is he speaking?*

### Burning Ropes

You have two ropes and some matches. The ropes burn irregularly like fuses when lit at either end. The first rope burns in e = 2.71828…hours and the second rope burns in √2 = 1.414213… hours.

*Produce a time interval as close as possible to 1 hour.*

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## Solutions to week 139

In Pondering Productivity, the hens in both farms are equally productive. Smith’s farm produces 245 eggs in 6 days; Jones’s farm produces 6,125 eggs in 24 days. In Cornfield Planning, cut out quarter circles on the corners with radius = 100/(2 + √π) = 26.5079… feet. This gives an area-to-perimeter ratio of 100/(2 + √π) = 26.5079…

**Pondering Productivity answer explained:**

(a) Eight of Smith’s hens will lay 7 × (7/6) eggs in 6/5 days for a rate of (7/8) × (7/6) ÷ (6/5) = 245/288 eggs per hen per day. Six of Jones’s hens will lay 5 × (7/6) eggs in 8/7 days for a rate of (5/6) × (7/6) ÷ (8/7) = 245/288 eggs per hen per day. The hens from both farms are equally productive.

(b) With 48 hens Smith gets 245/6 eggs daily so must wait 6 days to get 245 eggs.

(c) With 300 hens Jones gets 6125/24 eggs daily so must wait 24 days to get 6125 eggs.

**Cornfield Planning answer explained:**

Cut out quarter circles on the corners as shown in the figure. The area of such a shape is A = 10000 – 4r^{2} + πr^{2}. The perimeter of the shape is P = 400 – 8r + 2πr. The ratio A/P takes on a maximum value of 100/(2+√π) = 26.5079… when r = 100/(2+√π) = 26.5079… If the cornfield were either a circle or a square, the ratio A/P would be 25.

## Recent Weeks

**Week 139**: Pondering Productivity & Cornfield Planning, solutions to Close to a Quart & Math Party

**Week 138**: Close to a Quart & Math Party, solutions to Room for One More & End Points

**Week 137**: Room for One More & End Points, solutions to Sevens and Elevens & Three-Digit Squares

**Week 136**: Sevens and Elevens & Three-Digit Squares, solutions to Swimming Workout & Batting Order

**Week 135**: Swimming Workout & Batting Order, solutions to Maximization & Multiplication Table

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

**Come back next week** for answers and more puzzles.