This morning, Leaving Certs around the country sat down for Mathematics Paper 2. For many, this will have spelled the end of their formal education in maths. For others, it’s just beginning.
As some of those Leaving Cert students look towards futures studying maths or maths-related courses in college, the Irish Mathematical Trust (IMT) is looking to foster a love of maths among young children.
Providing hundreds of extracurricular lessons to thousands of students, engaging tens of thousands in mathematical activities through online support, and training the best students to mentor younger students, the IMT works to help young people to engage deeply with maths and understand its relevance.
It appears to be working. More than one-third of the 48 awards that Ireland has won over 30 years of participation in mathematical Olympiads have been won in the last three years by IMT students.
As those winning students no doubt know, to become proficient in maths, there are a number of skills that are of benefit. Prominent among these is an ability to think logically.
So this week – the first week of our new regular maths feature, with puzzles supplied by the IMT – we take a look at a problem of logic.
Backpacking through Lancre? Pack your thinking hat
When the tourist Twoflower arrived in the small mountain country of Lancre, it was late at night and everything was in complete darkness. The first inhabitants he met were five witches.
Having done his homework before travelling, Twoflower knew that each witch in Lancre wore either a grey or a pink dress. He also knew that the witches who wear grey always tell the truth, whereas the witches in pink always lie.
Trying to figure out who he was talking to – and who he could trust – Twoflower asked each of the witches what colour they were wearing.
- The first witch said, “I wear a grey dress”.
- The second witch said, “At least three of us wear a grey dress”.
- The third witch said, “The first witch wears a pink dress”.
- The fourth witch said, “At least three of us wear a pink dress”.
- The fifth witch said, “We all wear a pink dress”.
How many of the five witches wore a grey dress?
Scroll down for the solution to this week’s problem.
This week’s maths puzzle comes courtesy of Dr Bernd Kreussler – lecturer in Mathematics at MIC Limerick – who is actively involved in mathematics enrichment classes at UL and the Irish Mathematical Olympiad.
The answer is two.
After Twoflower heard each of the witches’ responses, he was able to figure out that two of the witches were wearing grey dresses. But how did he know?
First, he looked at the contrasting statements.
As we know, all the witches wear either a grey dress or a pink dress. The first witch – who claimed to be wearing grey – and the third witch cannot both be telling the truth, because the first witch can’t be wearing both. But they also can’t both be lying, because she also can’t be wearing neither.
Therefore, of the first witch and the third witch, we know that one is wearing grey (always tells the truth) and the other is wearing pink (always lies).
Knowing this, we know that the fifth witch – who said they were all wearing pink – didn’t tell the truth, so she’s wearing pink.*
We now know that two witches were wearing pink, and one was in grey.
Let’s now look at the responses from the second and fourth witches.
Because there are only five witches, we know it can’t be true that three are wearing grey and three are wearing pink. This means that they can’t both be telling the truth.
But neither can they both have lied. If they did, there would be only two in grey dresses and two in pink. As we know there are five witches, not four, we can deduce that one of them is telling the truth and the other is lying.
We therefore know that one is wearing grey, and the other pink.
This means that we know there are three witches in pink and two in grey, though – as we don’t know which of the first and third was lying – we can’t be sure who’s wearing what.
*Another way to see that the fifth witch didn’t tell the truth is because this would only be possible if she were wearing grey, contradicting her own response.
Main image via Shutterstock
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