Category Archives: Math Inquiries

Double Areas Puzzles

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A while ago James Tanton provided a series of puzzles:

Puzzle #1   At what value between 0 and 1 does a horizontal line at that height produce two regions of equal area as shown on the graph of y = x2?

Puzzle #2   A horizontal line is drawn between the lines y = 0 and y = 1, dividing the graph of y = x2 into two regions as shown. At what height should that line be drawn so that the sum of the areas of these two regions is minimal?

Puzzle #3   A horizontal line is drawn between the lines y = 0 and y = 1, dividing the graph of y = xn into two regions as shown (n > 0). At what height should that line be drawn so that the sum of the areas of these two regions is minimal? Does that height depend on the value of n?

Puzzle #4   What horizontal line drawn between y = 0 and y = 1 on the graph of y = 2√x – 1 minimizes the sum of the two shaded areas shown?

See Double Areas Puzzles for solutions.

Elliptic Circles

Here is another UKMT Senior Challenge problem for 2017.

“The diagram shows a square PQRS with edges of length 1, and four arcs, each of which is a quarter of a circle. Arc TRU has centre P; arc VPW has centre R; arc UV has centre S; and arc WT has centre Q.

What is the length of the perimeter of the shaded region?

A_6___B_(2√2 – 1)π___C_(√2 – 1/2)π ___D_2___E_(3√2 – 2)π”

Answer.

See Elliptic Circles for a solution.

Amazing Identity

This is a most surprising and amazing identity from the 1965 Polish Mathematical Olympiads.

“31.  Prove that if n is a natural number, then we have

(√2 – 1)n = √m – √(m – 1),

where m is a natural number.”

Here, natural numbers are 1, 2, 3, …

I found it to be quite challenging, as all the Polish Math Olympiad problems seem to be.

See the Amazing Identity

The Tired Messenger Problem

Here is another challenging problem from the Polish Mathematical Olympiads.  Its generality will cause more thought than for a simpler, specific problem.

“A cyclist sets off from point O and rides with constant velocity v along a rectilinear highway.  A messenger, who is at a distance a from point O and at a distance b from the highway, wants to deliver a letter to the cyclist.  What is the minimum velocity with which the messenger should run in order to attain his objective?”

See the Tired Messenger Problem

Incredible Trick Puzzle

Here is another typical sum puzzle from Presh Talwalkar.

“Solve the following sums:

_____1/(1×3) + 1/(3×5) + 1/(5×7) + 1/(7×9) + 1/(9×11) =

_____1/(4×7) + 1/(7×10) + 1/(10×13) + 1/(13×16) =

_____1/(2×7) + 1/(7×12) + 1/(12×17) + … =”

The only reason I am including this puzzle is that Talwalkar gets very excited about deriving a formula that can solve sums of this type.  This gives me an opportunity to discuss the “formula vs. procedure” way of doing math.

Answer.

See the Incredible Trick Puzzle for solutions.

Fireworks Rocket

This is another physics-based problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) that may take a bit more thought.

“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.

Assuming no air resistance and taking g = 9.8 m/s2, how long does it take to reach its maximum height?”

I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.

Answer.

See the Fireworks Rocket for solutions.

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