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King's Maths School

Challenge 452: A Party Trick

Amaze your friends with this trick! (Maybe, if they like maths...) Thanks to KCLMS student Arthur for the problem.

(1) Take any monic quadratic polynomial, i.e. an expression of the form x2 + bx + c.

Pick any three consecutive integers, e.g. 4, 5, 6 or -9, -8, -7.

Substitute your largest and smallest integers into your quadratic and add the results. Call this A.

Then substitute your middle integer into your polynomial and double the result. Call this B.

Finally, evaluate A - B. What do you notice? Can you prove that it always works?

(2) Can you adapt the trick to work with a monic cubic polynomial, i.e. ax3 + bx2 + cx + d?

Can you generalise to a polynomial of any degree?

Submit your solution

Please do send in your solution to this problem to weeklymaths@kcl.ac.uk You can scan or photograph your written work, or type your solutions. If this is your first weekly maths challenge solution, please include your year group and the name of the school you attend. We'll be happy to provide feedback on your solution, assuming that you are in year 11 or below. If you are older than this, we hope you enjoy trying the problems and reviewing your solutions against those we publish on the website.