MATHS WEEK STARTED yesterday and, as is our annual tradition, we’re setting our readers some puzzles. Give them a go!

Day Two - Happy Birthday Quaternions

Of course all the brother’s sums isn’t done in the digs. He does be inside in a house in Merrion Square doin’ sums as well. If anybody calls, says the brother, tell them I’m above in Merrion Square workin at the quateernyuns, says he, and take any message. There does be other lads in the same house doin’ sums with the brother. The brother does be teachin’ them sums. He does be puttin’ them right about the sums and the quateernyuns.

Myles na gCopaleen, Cruiskeen Lawn

16 October is the birthday of quaternions. Quaternions are sets of four numbers that can be used to describe the position and orientation of bodies in space. They are useful for moving objects in computer animation and manoeuvring spacecraft.

They were discovered on this day in 1843, by William Rowan Hamilton while walking along the towpath of the Royal Canal at Broombridge. The occasion and location is marked with a plaque and the birthday is celebrated by the Irish mathematical community with the annual Hamilton walk.

For Myles, quaternions represented advanced mathematics impenetrable to all but the experts at the Dublin Institute for Advanced Studies (in Merrion Square) and of course “the brother”.

You will be relieved that our puzzles today are about birthdays, not quaternions.

1. John and Mary celebrate their birthdays on the same day. Mary is now four times John’s age. In four years she will be twice John’s age. How old is Mary now?

2. When I asked her how old she was, my friend smiled and said cryptically, “The day before yesterday I was twenty-two, but next year I’ll be 25.

When was her birthday, and on what date did the conversation take place?

Come back tomorrow for the answers to today’s puzzle.

The Maths Week puzzles this year are presented by Colm Mulcahy, professor emeritus of Mathematics at Spelman College, USA, and adjunct professor with Calmast at South East Technological University. Colm is chairperson of the Martin Gardner Foundation USA, and the curator of the website mathsireland.ie

Saturday’s puzzle: The answer 

A. You actually end up earning 1% less that you did originally. For every €100 you earn at
the outset, after the 10% raise you earn €110, but then after the later 10% reduction
that drops €11 to only €99. That’s a 1% loss overall.

B. Your brother actually ends up earning the same as he did originally. For every €100 he
earns at the outset, after the 25% raise he earns €125, but then after the later 20%
reduction that drops back €25 to the original €100. That’s because 20% of €125 is a fifth
of it, which is €25.

C. Your sister actually end up earning 2% more that she did originally. For every €100 she earns at the outset, after the 20% raise she earns €120, but then after the later 15% reduction that drops €18 to only €102. That’s because 15/100 times €120 is €18.

D. Your friend actually end up earning 5% less that she did originally. This is very surprising. For every €100 she earns at the outset, after the 25% raise she earns €125, but then after the later 24% reduction that drops €30 to only €95. That’s a 5% loss overall. Note that 24/10 times €125 is €30.

E. The overall pay cut is a little over 67%: for every €100 you earn at the outset, you end
up earning about €32.77. Can you explain why, using the kind of reasoning we used
above?

Say you start with €100…

• 20% of €100 is €20 so you reduce to €100 – €20 = €80
• 20% of €80 is €16 so you reduce to €80 – €16 = €64
• 20% of €64 is €12.80 so you reduce to €64 – €12.80 = €51.20
• 20% of €51.20 = €10.24 so you reduce to €51.20 – €10.24 = €40.96
• 20% of €40.96 = €8.19 so you reduce to €32.77

That’s a total loss of €67.33 from the original €100 , so overall, a reduction of about 67%.

Moral: percentages are tricky! They don’t work like ordinary addition and subtraction: what gets added and subtracted to modify amounts repeatedly involves several multiplications. The final results are not easy to predict. As we have just seen, it’s even hard to predict qualitatively: will we come out ahead, even, or behind?