Sometimes, maths is a matter of numbers. Sometimes, it’s all about letters. And, sometimes, it just involves logic.
The future is automated. That’s what the technologists keep telling us, anyway.
We’re going to have robots for everything, from car manufacture to medicine, to journalism (eep!). We’ll probably even have robot security guards that can see absolutely everything.
But what good are robot security guards if they’re capable of lying about what they see?
At some indeterminate point in our future, the company XYZ will start making security robots called OLYs, or Observational Law-abiding Yeoman (language will take a little retroactive turn between now and then).
OLYs have 10 ‘eyes’ – all the better to see crime with – enabling the robots to see everything, except their own eyes.
When an XYZ-brand robot is functioning normally, it is completely truthful and accurate. Every now and then, though, a defect XYZ robot is created. The defect robots have fewer than 10 eyes, and they always lie.
The XYZ robots OLY1, OLY2, OLY3, OLY4 and OLY5 are gathered by themselves in a room.
Having spent some time getting to know each other, they start shooting the breeze. Being a generally judgemental bunch, XYZ-brand robots are very dismissive of defect robots. They’re also big fans of maths. So a first step upon meeting a new OLY is to count its eyes, then discuss it with everyone else in the room.
After each counts the others’ eyes, OLY1 claims to see 40 eyes, OLY2 sees 39, OLY3 sees 38, OLY4 sees 37 and OLY5 sees 36.
Can you determine which of the OLYs are functioning normally and which are defects?
Scroll down for the solution to this week’s puzzle.
This week’s maths puzzle comes courtesy of Dr Anca Mustata, lecturer in Mathematics at UCC, who is actively involved in the Maths Circles initiative, the Mathematics Enrichment programme in UCC and the Irish Mathematical Olympiad.
All are defect!
If OLY1 was telling the truth, then all of the other robots would have 10 eyes each. In that case, they would also all tell the truth and would, therefore, all be claiming to see 40 eyes. As they all have different claims, we know OLY1 must be lying.
This means that OLY2 can see, at most, 39 eyes (10+10+10+9), which is what it claims to see. If OLY2 was speaking the truth, however, all the other OLYs would also have 10 eyes, and would also claim to see 39 eyes in the room. Therefore, OLY2 is lying, and must also be defect.
Since OLY1 and OLY2 have, at most, nine eyes each, then OLY3 can see, at most, 38 eyes, which it indeed claims to do. But, as with OLY1 and OLY2, the other bots let OLY3 down. As they all see different numbers, OLY3 must also be lying and, therefore, is defect.
Following this line of reasoning, we see that only OLY5 could be telling the truth. If the other four OLYs were defect, OLY5 would, indeed, see 36 eyes (9+9+9+9).
However, if OLY5 were telling the truth, that would mean that OLY4 was also correct, seeing 37 eyes (9+9+9+10). As this is a tautological impossibility, rife with internal contradictions, we must therefore assume that OLY5 also lied and is also defect.
Main image via Shutterstock