## ________________

With the new year approaching, the passage of time is on people’s minds.

## ________________

### Breakfast, Lunch, or Dinner?

Just as team member Malcolm is finishing a meal, he notices the minute hand is on a minute mark exactly one minute ahead of the hour hand.

*Which meal is Malcolm finishing?*

### The 2018 Puzzle

The coach has an interesting problem to celebrate the coming year.

*“You have 2018 distinct points in space. Prove that there is a plane that divides the points into two sets, with 1009 points on each side of the plane.”*

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## Solutions to week 119

In Solar Cap Versus Polar Cap, the solar cap and polar cap have equal areas. The pole in How Tall Is the Pole is ≈ 180.75 feet tall.

**Solar Cap Versus Polar Cap answer explained:**

Let the compass radius be r. On a sphere of radius R the area of a spherical cap of half-angle θ is 2π(1 – cosθ)R^{2} (θ is shown in the figure.) Also from the figure, cosθ = 1 – 2sin^{2}(θ/2) = 1 – 2r^{2}/(2R)^{2}. Putting these together gives the area of the spherical cap as πr^{2}, independent of R.

**How Tall Is the Pole answer explained:**

Let h be the height of the pole. r = 20996000 feet is the earth’s radius of curvature, and d is the extra length of chain needed to accommodate the pole. Then, from the figure,

d = 2r(tanθ – θ) = 2r(θ^{3}/3 + 2θ^{5}/15 + …) ≈ 2rθ^{3}/3

h = r(secθ – 1) ≈ rθ^{2}/2 = (9d^{2}r/32)^{1/3} ≈ 180.749… feet.

Another way to solve the problem is to use elementary trigonometry and the trigonometric equation solver found here: http://www.wolframalpha.com/widgets/view.jsp?id=27b8624af4f632acc36d65c4a7eb715.

## Recent Weeks

**Week 119**: Solar Cap Versus Polar Cap & How Tall is the Pole, solutions to Relations Puzzle & Square Purchases

**Week 118**: Relations Puzzle & Square Purchases, solutions to Fifty-Fifty & Random Toss

**Week 117**: Fifty-Fifty & Random Toss, solutions to Cutting the Domino & Divide by Three

**Week 116**: Cutting the Domino & Divide by Three, solutions to Thanksgiving Split & Easy as Pie

**Week 115**: Thanksgiving Split & Easy as Pie, solutions to Primes and Products & Two Pints of Cider

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

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