Tag Archives: algebra

Consecutive Product Square

This problem from Colin Hughes at Maths Challenge is a most surprising result that takes a bit of tinkering to solve.

“Problem
We can see that 3 x 4 x 5 x 6 = 360 = 19² – 1. Prove that the product of four consecutive integers is always one less than a perfect square.”

The result is so mysterious at first that you begin to understand why the ancient Pythagoreans had a mystical relationship with mathematics.

See the Consecutive Product Square.

(Update 11/12/2020) Generalization and Visual Proof
Continue reading →

Cube Roots Problem

In a June Chalkdust book review of Daniel Griller’s second book, Problem solving in GCSE mathematics, Matthew Scroggs presented the following problem #65 from the book (without a solution):
“Solve _______________”

Scroggs’s initial reaction to the problem was “it took me a while to realise that I even knew how to solve it.”

Mind you, according to Wikipedia, “GCSEs [General Certificate of Secondary Education] were introduced in 1988 [in the UK] to establish a national qualification for those who decided to leave school at 16, without pursuing further academic study towards qualifications such as A-Levels or university degrees.” My personal feeling is that any student who could solve this problem should be encouraged to continue their education with a possible major in a STEM field.

Answer.

See Cube Roots Problem for a solution.

Radical Radicals

This is another UKMT Senior Challenge problem, but for the year 2005. I thought it was diabolical and hadn’t a clue how to solve it. Even after reading the solution, I don’t think I could have come up with it. I take my hat off to anyone who solves it.

“Which of the following is equal to

Answer.

See Radical Radicals for a solution.

Challenging Sum

Here is a problem from the UKMT Senior (17-18 year-old) Mathematics Challenge for 2009:

“Four positive integers a, b, c, and d are such that

abcd + abc + bcd + cda + dab + ab + bc + cd + da + ac + bd + a + b + c + d = 2009.

What is the value of a + b + c + d?

A 73_________B 75_________C 77_________D 79_________E 81”

Answer.

See the Challenging Sum for a solution.

(Update 4/17/2019) Continue reading →

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