This month, I was able to use the sentence “Don’t
worry,I’mamathematician”.Ihaveneverbeenina
suchasuper-positionofdelightandembarrassment.
The backyard of my house is on a serious gradient,
andI’mkeenforitnottogetanyideasaboutbecoming
thefrontyard.SoIhadsomeonearoundtogivemea
quote on how much holding back the gravitational
potential energy could potentially cost. They took
some measurements, drew some diagrams and then
expressedregretthattheycouldn’tmeasuretheexact
heightofthebackoftheyard.It’dbenicetoknowthis,
they said, but getting a surveyor in would be too costly.
Which is where we came in: some string, a home-
made inclinometer and a bit of trigonometry later, I
emailed them the exact height and gradient of the
yard. It seems landscaping calculations are more
complex,becauseI’venotheardbackfromthemyet.
MyjoyatusingPythagorasandtrigratiosina
practical situation reveals the shameful secret of
maths:mostofthecontentwillbeoflittletozerouse
in your normal life. This case was an anomaly, an
enjoyableoutlier.LikemanypeopleIenjoyeddoing
calculations with angles and side-lengths in school;
like sudoku, but with more satisfying chains of logic.
NowIlookenviouslyatmyphoneasiteffortlessly
crunches through GPS calculations, doing all the fun
work and delivering only the final answer to me.PUZZLESSo I’ve decided to forget waiting for the chance to
calculate a side-length to crop up and I’ve sought out
some Pythagorean puzzles. Like me, you can embrace
the fun of geometric gymnastics!
The first puzzle is my favourite, and relies on a bit of
Pythagoras flexibility. You have four identical circles
arrangedinasquarejusttouchingeachother.This
produces two types of spaces: the inside of the circles
and the region between all four circles. We now put in
two new small circles: Circle A is the biggest circle of
which four, arranged as a square, can fit inside one of
theoriginalcircles;CircleBisthebiggestcirclewhich
canfitbetweentheoriginalcircles.Thechallengeis
simple:whichisbigger,CircleAorCircleB?
Thesecondpuzzleisnicebecauseit’sbrandnew.
Some people will have seen the first puzzle before, but
this second one was created in 2016 by Canadian
maths enthusiast John Barsby (and member of the
Winnipeg MathsJam). So unless you competed in the
2016 Manitoba Mathematical Contest it was originally
written for, or came across it when it spread through
the MathsJam, this should be all new.
Circle1andCircle2haveradii1and2respectively,
andareplacedinasnowmanformation.Atriangleis
drawn around them, fitting as snugly as possible. What
we want to know is: how big are the two identical
circles that fit in at the bottom?Matt Parkerisastand-upcomedianandmathematician.Heis
aregularonBBCRadio4’sThe Infinite Monkey Cage, alongside
Brian Cox and Robin Ince.THE NUMBER GAMES
GIVE YOUR BRAIN A WORKOUT
WHAT
ISMATHSJAM?
MathsJam is a chance for
maths lovers to get together in a
pub and swap puzzles over a
drink. Matt organises the London
and Guildford MathsJams, and
colleague Katie runs the
Manchester one. Visit
mathsjam.comto find your
closest gathering.Space between the circles Which is bigger: A or B?Four circles on a square Space inside a circleBAr = 1r = 2r =?PUZZLE 1 PUZZLE 2