MATHS WEEK IS under way and, as is our annual tradition, we’re setting our readers some puzzles. Give them a go!
Happy Broomsday!
Ireland’s greatest mathematician, William Rowan Hamilton, was born in Dominick Street Dublin in 1805 and it is claimed that he knew a dozen languages at the age of 10.
He went to study in Trinity College Dublin, where he showed great ability and was chosen for the position as the Andrews Professor of Astronomy there before he graduated.
With this came the title of Royal Astronomer for Ireland and a residence at Dunsink Observatory to the West of the city. On the 16 October 1843 he set out to walk with his wife from Dunsink into the city to a meeting of the Royal Irish Academy.
While he was passing Broom Bridge in Cabra, the solution to a problem he had been working on for years came to him in a flash of inspiration. He carved the strange equation for quaternions on the bridge.
A plaque at the site bears the inscription:
He later wrote: “I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i,j,k; exactly such as I have used them ever since.”
This strange equation was the foundation for a new form of algebra with new rules. It revolutionised 19th-century mathematics. Another new algebra, Boolean Algebra was to come shortly after.
Although influential, the quaternions didn’t have any significant applications until the late 19th Century, when they were found to be ideal for representing rotation in three dimensions and very useful for controlling the orientation and position of spacecraft and robots and objects in computer generated graphics.
The 16 October has been dubbed “Broomsday” and is celebrated in the annual Hamilton Walk and RIA Hamilton Lecture, both of which will take place online today.
Why is it useful to have a mathematics to describe rotation in three-dimensional space?
If you move three paces forward and three paces to the right, you will end up in the same place as if you move three paces to the right and then move three paces forward. And, you will be facing the same way.
This movement along the ground can be drawn out on a map or chart with movement along in an “x” and a “y” direction. The movement can also be described by algebra. It all gets more complicated when you try to go to three dimensions and include rotation.
Hamilton was looking for a way that could describe this motion and rotation in three dimensions and he worked on the problem for years.
It was such a pre-occupation that his sons would ask him, “Well, Papa, can you multiply
triplets?” and poor Hamilton, would reply with a sad shake of the head: “No, I can only add and subtract them.”
Your challenge today is to explore this movement.
Imagine you have a normal die (opposite sides on dice add up to 7). Imagine you place it before you on a table with the number one facing up, two facing towards you and three facing to your right.
If you rotate the die a quarter turn (90 degrees) away from you, the two will be facing up. Rotate it again two more quarter turns away from you.
Then rotate it two quarter times to your right. What number is facing up now?
Repeat this exercise with the same starting conditions, but this time rotate the die two quarter turns to your right first and then three quarter turns away from you. Your die will be end up in the same finishing place, but what number is now facing up?
All puzzles are by Eoin Gill, the coordinator of Maths Week Ireland and director Calmast STEM Engagement Centre, Waterford Institute of Technology
Come back tomorrow for Saturday’s puzzle and the answer for today’s.
Thursday’s puzzle: The Answer
12 cows and 4 milking staff
The Method:
First let us assemble all the information we have been given, and we can deduce: We have staff, stools and cows that have legs (2, 3 and 4 respectively).
Each staff member and the cows have one head each.
Let us call the number of staff, MM. With two legs each the number of their legs will be 2MM.
The number of cows will be called C. The number of cows’ legs will be 4C.
The number of milking-stools will be called MS. The number of stool legs will be 3MS.
Then we can express the total number of legs as follows:
2MM + 4C + 3MS = 68
As cows and staff will have one head each, the total number of heads will be:
MM + C = 16
We can deduce that there should be one stool per staff member.
So MS = MM
So we can write:
2MM + 4C + 3MM = 68
Which is:
5MM + 4C = 68
But if MM + C = 16 we can write this as:
C = 16 – MM
And replace the C in the “legs” equation with this.
5MM + 4(16-MM) = 68
5MM + 64 -4MM = 68
MM = 68 – 64
MM = 4
And since MM + C = 16, then C = 16 – 4 = 12
Hence, 12 cows and 4 staff
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