Irrational – Miles Writes
I was sorry that much of the reaction to Irrational concerned the ambiguity surrounding the C and F values. In retrospect this could have been avoided by clarifying in the preamble that √C was a surd and that F had to be a positive integer.
I had assumed that expanding the brackets using (A + B√C)² = A² + B²C + 2AB√C would mean that E = A² + B²C and F =2AB. I now realise that when C = 1, 4 or 9 other options for E and F are possible. I must admit that this came as quite a shock to me!
The question arises as to whether other answers to the puzzle, subject to constraints, are feasible, and shockingly this is so! For instance at 14d the intended combination was (8 + 7√5)³ = 6392 + 3059√5, but (8 + 7√1)³ = 6392 – 3017√1, (8 +7√4)³ = 6392 + 2128√4 and even (8 + 7√9)³ = 6392 + 5999√9 are all possible. I note from the results that 86 solvers were marked correct whereas 2 were marked wrong, which means two things: (1) most solvers were not put off after all and (2) two people might request a recount.
As setters know there is often a long time between an initial concept and the eventual publication, and I think that my original title was A + B(SURD), thereby clarifying what C could and could not be, but rejected this when the highlighting of π became a possibility and opted for Irrational instead.
That said, there are other comments I would like to make. I do not think that using the Binomial Theorem was obligatory, and the intention was that solvers only needed to multiply out brackets using the Distributive Law, which (eventually) the editors agreed was within GCSE Maths guidelines. And my impression has always been that most Magpie solvers relish a challenge and, with the hint of a ‘final step’, they might persevere through the obstacles to expect a PDM at the end. Also the editors felt that Irrational had similar features to Root 66 in the Listener, but I maintained that it had sufficient stand-alone qualities to merit an airing. But most importantly I quite enjoy exploring mathematical diversions, getting quite excited by revelations and/or discoveries and like to share these with others. Over the years I am pleased that The Magpie has supported my self-indulgence.
August 2nd, 2024 at 5:20 pm
Miles: Many thanks for the blog which is much appreciated. You have my sympathies regarding the preamble. Why the editors didn’t go with the full clarification of all the letters as in The Listener puzzle is a mystery to me?
Their reason for not putting out a clarification is somewhat reminiscent of the Sturgeonesque dismissive wave of the hand that she used all too often!
I agree you didn’t need to use the Binomial Theorem, it just made it quicker if you did. Alternatively you could have used WolframAlpha to work things out which I suspect some solvers did.
I note that it’s not going to put you off setting more which is good. Mike Rich rejected the first puzzle I set but ended his letter with “Don’t give up”. I didn’t!
Oyler
PS Thanks for Chambers Dictionary