IT’S HUMP DAY of Maths Week, so take a break and see if you can solve this one.
Ireland says farewell to Jan Łukasiewicz (1878 – 1956)
It can be argued that we in Ireland pay little heed to our mathematical and scientific heritage.
In central and eastern European countries however, mathematics is held in high regard.
On 22 November, a delegation from Poland, including the Polish Air Force, will visit Ireland to repatriate the remains of Jan Łukasiewicz. They will take him home to be re-buried in his native land.
Łukasiewicz has rested in Mount Jerome Cemetery in relative obscurity since 1956 and is almost unknown in this country – but is widely celebrated in Poland.
Łukasiewicz was a philosopher who made important contributions to mathematics and logic. He was born in 1878 and became a senior figure in Polish academic life, becoming rector of the University of Warsaw and even minister for education for a time.
He married Regina Barwińska in 1928 and towards the end of the Second World War, he moved with his family to Germany to escape the Soviet advance.
After the war, he was made welcome in Ireland and served as a professor in UCD until his death in 1956.
Among Łukasiewicz’s many contributions to knowledge was the invention Polish Notation in 1924.
Our standard operators in maths are addition, subtraction, multiplication and division. We have an agreed system of writing mathematical expressions so that everyone can understand each other. Our normal system is to place the operators between the numbers or symbols.
For instance 2 + 3 x 4.
This however could be interpreted as add 2 to 3 and then multiply by 4 which will give 20. Someone else might multiply 3 by 4 and then add 2, which will give 14.
In order to avoid these problems, we have agreed rules for the order in which we do the operations. Multiplication is performed before addition, so we should all get the answer 14. To avoid ambiguity in complex expressions, we use brackets.
Łukasiewicz’s notation places the operators in front of the operands (symbols or numbers). It seems strange that someone would want to do this, but his system was later found to be very useful in computing.
And so, while most of us have been blissfully unaware of the influence of Łukasiewicz’s work in all our lives, his remains have rested in Dublin’s Mount Jerome Cemetery.
In November, we will say, “Thank you and farewell Jan.”
A guided tour of Mt Jerome Cemetery – lead by Colm Mulcahy – to the resting place of many of Ireland’s greatest mathematicians and scientists will take place Saturday next, 22 October at 1.30pm. Places are limited and booking is essential at www.mathsweek.ie.
Our challenge today is to explore Polish Notation (here we will use * as the multiplication symbol and / as the division symbol).
Our expression above 2+3*4 becomes +2 * 3 4 in Polish Notation and it works as follows:
*2 3 is 2*3
2*3+4 is (2*3)+4 = 10 and that becomes +* 2 3 4 in Polish Notation.
(We evaluate *2 3 first = 6 and then +4 6 = 10)
-+*2 3 4 5 (2*3 first, then 6+4, then 10-5 = 5)
1. The following expressions are in Polish Notation, can you work out the answers?
A. – 10 5
B. +- 6 5 *4 3
C. –*+ 5 3 2 10
D. /+*2 4 *4 5 2
2. Convert the following expressions into Polish Notation:
A. 2 * (3 + 4)
B. (2 * 3) + (4 *5)
C. 6 + 5 – 4 + 3
D. 6*5/(7+3)
Come back tomorrow for the answers to today’s puzzle.
Tuesday’s puzzle: the answer
There is no hope if the two numbers we are using share a common factor.
For example, with 2 and 6 (which have a common factor of 2) we can never get any odd totals, only even ones, namely multiples of 2. With 6 and 15 we will only get multiples of the common factor 3 (leaving out lots of totals, including 10, 100, 1000, etc).
Now, consider two numbers with no common factor. It turns out that there’s a simple way to find the tipping point. For the first numbers 3 and 5 in the puzzle it is 8.
For 5 and 7, a lot of trial and error shows that it’s 24. For 4 and 9 it’s also 24. For 6 and 7 it’s 30. What’s going on here?
The way to find the tipping number for two numbers which don’t share a common factor is subtract their sum from their product, and add 1.
So for 5 and 7 we subtract 12 = 5 + 7 from 35 = 5 * 7, getting 35 – 12 = 23, then add 1, to get 24. Check the other examples and try it for other suitable pairs of numbers.
Summarising, for numbers X and Y, the tipping point is XY – X – Y +1.
If three coins of specific different values are used, we can once again experiment to find the tipping points each time. But amazingly nobody has ever found a formula like the one for pairs of values above. And it’s not for the want of trying: there is good reason to believe that no such formula exists. Welcome to the weird and wonderful world of mathematics!
Maths Week Ireland is coordinated by SETU with partners across the island of Ireland. This year over 400,000, north and south will take part and these puzzles give you a chance to participate.
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
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