Category Archives: Math Inquiries

Pairwise Products

This 2005 four-star problem from Colin Hughes at Maths Challenge is also a bit challenging.

Problem
For any set of real numbers, R = {x, y, z}, let sum of pairwise products,
________________S = xy + xz + yz.
Given that x + y + z = 1, prove that S ≤ 1/3.”

Again, I took a different approach from Maths Challenge, whose solution began with an unexplained premise.

See the Pairwise Products

Flipping Parabolas

This is a stimulating problem from the UKMT Senior Math Challenge for 2017. The additional problem “for investigation” is particularly challenging. (I have edited the problem slightly for clarity.)

“The parabola with equation y = x² is reflected about the line with equation y = x + 2. Which of the following is the equation of the reflected parabola?

A_x = y² + 4y + 2_____B_x = y² + 4y – 2_____C_x = y² – 4y + 2
D_x = y² – 4y – 2_____E_x = y² + 2

For investigation: Find the coordinates of the point that is obtained when the point with coordinates (x, y) is reflected about the line with equation y = mx + b.”

See Flipping Parabolas.

Fibonacci, Chickens, and Proportions

There is the famous chicken and the egg problem: If a chicken and a half can lay an egg and a half in a day and a half, how many eggs can three chickens lay in three days? Fibonacci 800 years ago in his book Liber Abaci (1202 AD) did not have exactly this problem (as far as I could find), but he posed its equivalent. And most likely the problem came even earlier from the Arabs. So we can essentially claim Fibonacci (or the Arabs) as the father of the chicken and egg problem. Here are three of Fibonacci’s actual problems:

  1. “Five horses eat 6 sestari of barley in 9 days; it is sought by the same rule how many days will it take ten horses to eat 16 sestari.
  2. A certain king sent indeed 30 men to plant trees in a certain plantation where they planted 1000 trees in 9 days, and it is sought how many days it will take for 36 men to plant 4400 trees.
  3. Five men eat 4 modia of corn in one month, namely in 30 days. Whence another 7 men seek to know by the same rule how many modia will suffice for the same 30 days.”

By modern standards these problems all involve simple arithmetic to solve. But there are actually some subtleties in mapping the mathematical model to the situation, in which fractions, proportions, ratios, and “direct variation” get swirled into the mix—naturally causing some confusion.

See Fibonacci, Chickens, and Proportions

Maximum Product

This 2007 four-star problem from Colin Hughes at Maths Challenge is definitely a bit challenging.

Problem
For any positive integer, k, let Sk = {x1, x2, … , xn} be the set of [non-negative] real numbers for which x1 + x2 + … + xn = k and P = x1 x2 … xn is maximised. For example, when k = 10, the set {2, 3, 5} would give P = 30 and the set {2.2, 2.4, 2.5, 2.9} would give P = 38.25. In fact, S10 = {2.5, 2.5, 2.5, 2.5}, for which P = 39.0625.

Prove that P is maximised when all the elements of S are equal in value and rational.”

I took a different approach from Maths Challenge, but for me, it did not rely on remembering a somewhat obscure formula. (I don’t remember formulas well at my age—only procedures, processes, or proofs, which is ironic, since at a younger age it was just the opposite.) It is also clear from the Maths Challenge solution that the numbers were assumed to be non-negative.

See Maximum Product.

Magic Pythagorean Circle

This statement showed up recently at Futility Closet and I found it to be another one of those magical results that seemed so surprising. I don’t recall ever seeing this before.

“The radius of a circle inscribed in a 3-4-5 triangle is 1.
(In fact, the inradius of any Pythagorean triangle is an integer.)”

(A Pythagorean triangle is a right triangle whose sides form a Pythagorean triple.) Futility Closet left these remarkable statements unproven, so naturally I felt I had to provide a proof.

See Magic Pythagorean Circle

Alberti’s Perspective Construction

I was reading yet another book on the Scientific Revolution when I came across a discussion of the mathematical significance of the invention of perspective for painting in the 15th century Italian Renaissance. The main player in the saga was Leon Battista Alberti (1404 – 1472) and his tome De Pictura (On Painting) (1435-6), which contained the first mathematical presentation of perspective. Even though mathematics was advertised, it was not at the level of trigonometry I used in my post “The Perspective Map”, but rather entailed simple Euclidean plane geometry. So the discussion was largely historical rather than mathematical. Nevertheless, I became curious to learn how much Alberti was able to discover about perspective without a lot of math. This essay is the result.

See Alberti’s Perspective Construction

(Update 7/29/2019)  I got a response! Continue reading