Tag Archives: algebra

Movie Projector Problem

Here is another Brain Bogglers problem from 1987.

“Exactly four minutes after starting to run—when the take-up reel was rotating one and a half times as fast as the projecting reel—the film broke. (The hub diameter of the smaller take-up reel is 8 cm and the hub diameter of the projecting reel is 12 cm.) How many minutes of film remain to be shown?”

This feels like another problem where there is insufficient information to solve it, and that makes it fun and challenging. In fact, I was stumped for a while until I noticed something that was the key to completing the solution.

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1770 Card Game Problem

This problem from the 1987 Discover magazine’s Brain Bogglers by Michael Stueben apparently traces back to 1770, though the exact reference is not given.

“Here’s an arithmetic problem taken from a textbook published in Germany in 1770. Three people are gambling. In the first game, Player A loses to each of the others as much money as each of them had when the game started. In the next game, B loses to each of the others as much money as each had when that game began. In the third game, A and B each win from C as much money as each had at the start of that game. The players now find that each has the same sum, 24 guineas. How much money did each have when play began?”

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Consecutive Product Square

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

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.

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.

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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”

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(Update 4/17/2019) Continue reading