Varsity Math, Week 150

Varsity Math 150

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Welcome to Varsity Math, the weekly math puzzle column by the National Museum of Mathematics and featured each weekend in the Wall Street Journal.

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29

Make 29

Produce a mathematical expression equaling 29 in which you use the numeral 2 twice, 6 once, and +, -, ×, ÷, decimal points, and parentheses as many times as you wish.

Circle of Friends

The math team members position themselves in a circle. Each is given a number and shows it to his or her two neighbors. Each then takes the average of the two numbers he or she was shown and announces the result. The announced numbers turn out to be 1, 2, 3, … and so on all around the circle in order. Cindy exclaims, “The number I was given initially is ten times the number I announced!”

How many members are in the circle if there are fewer than 30 in the group?

momathlogohorizontalWSJLogo

Solutions to week 149

The solution to Tile the Square is shown in the figure below. In Largest Circle, the radius of the largest circle that can be drawn inside the polygon is r = (√3 – √2 + 1)/2 = .6589186…

Tile the Square

Largest Circle answer explained:
Draw the triangles ABC and A’B’C as shown. The largest circle within the polygon will be the smaller of the two inscribed circles of ABC and A’B’C. In ABC let c = AB = 1 + √3.

Then b = AC = sin135°/sin15°, a = sin30°/sin15° and the radius of the inscribed circle is r = .5 × (a + b – c)tan7.5° = (√3 – √2 + 1)/2 = .6589186…

In A’B’C let c’ = A’B’ = 2. Then b’ = A’C = c’sin45°/sin15°, a’ = c’sin120°/sin15° and the radius of the inscribed circle is r’ = .5 × (a’ + b’ – c’)tan7.5° = .6685466…

The smaller of the two radii is r = (√3 – √2 + 1)/2 = .6589186…

Largest Circle

Recent Weeks

Week 149: Tile the Square & Largest Circle, solutions to Cats and Dogs & Sporting Question

Week 148: Cats and Dogs & Sporting Question, solutions to Alternative Sum and Product & Math Quiz

Week 147: Alternative Sum and Product & Math Quiz, solutions to Mailing Problem & Pool Party

Week 146: Mailing Problem & Pool Party, solutions to Find m and n & Escape Room

Week 145: Find m and n & Escape Room, solutions to Liar’s Club & Palindrome Times

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

Come back next week for answers and more puzzles.