MATHS WEEK STARTED on Saturday and, as is our annual tradition, we’re setting our readers some puzzles. Give them a go!
Day 8: Change your point of view
“Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.” - George Poyla, How To Solve It (1945)
This ends our series of puzzles for Maths Week Ireland 2021. Hopefully you will have tried the challenges this week and maybe you will have had a taste of Poyla’s “triumph of discovery”.
More importantly though, I hope you learned some strategies to tackle future problems. We have selected problems that are instructional, that relied on mathematical thinking rather than mathematical knowledge.
The purpose was to show that there are strategies that are useful in approaching many problems and that they can be learned.
This week we have looked at some of them:
You may have used other strategies.
If you struggled, don’t despair.
Mathematical thinking is not hardwired – like other physical or mental abilities, it can be developed. Don’t be afraid of making mistakes – think of them as signposts for learning.
The key is practice and reflection.
As you do problems, reflect on what worked and what didn’t; ask yourself how you might have done it differently or better, and importantly, think would these strategies work in other cases?
In this way, you will build up an armoury of problem solving strategies that will increase your problem solving powers. Search for puzzles at your current level, learn from them, but also start to push yourself.
We finish the week with a selection of puzzles to exercise your freshly honed problem solving skills.
One
A car travels 24,000km. Six tyres were worn out over this distance. The car owner ensured that each tyre travelled the same distance as the others.
The car has four wheels. Tyres being in the boot don’t count: we are counting the distance for which they are in contact with the road.
How far did each tyre travel?
Two
Lord Fizzleworth hires three people from the village to wait at his annual garden party. He pays them very poorly, so when they’re clearing up afterwards they decide to help themselves to the excellent wine left over.
There are 7 full bottles, 5 bottles half full and 3 empty bottles. They know that the unscrupulous innkeeper will pay them €10 for a full bottle, nothing for a half full bottle, except the 50c deposit refund for an empty bottle.
How should they divide up their spoils so that each person gets an equal share? Also, how much could that be?
Three
The cold water tap (on its own) will fill the bath up to the overflow mark in 12 minutes. The hot water tap (on its own) will fill it in 15 minutes.
When the taps are off and the plug is pulled the bath will empty (at a constant rate) in 6 minutes.
The two taps are turned on full and the bath fills up to the overflow level. The plug is then pulled with the taps left running.
How long will it take for the bath to empty?
Four
A swimmer swims one kilometre up river, turns around and swims back to where she started. She maintains the same effort throughout; it takes her one and a half hours to cover the kilometre going upstream but only a half an hour to get back.
If she was swimming with the same effort with no current, say in a lake, how long would it take her to swim one kilometre?
(Scroll for the answers)
Puzzles compiled for The Journal by Eoin Gill of Maths Week Ireland / Waterford Institute of Technology.
The answers to Friday’s puzzles
1. Classic Car puzzle
Yes, it was probably too easy, but just a bit of fun. All you needed to do was look at the car parking spaces ‘upside down’. The answer is 87.
2. Train puzzle
The trains are 100km apart. One is travelling at 40kmph and one at 60kmph, therefore the gap is closing at 100kmph. So it will be 1 hour to the crash. Therefore, we just have to calculate how far the fly can fly in 1 hour. Which is 200km.
3. Croagh Patrick
This one requires a clever insight. What if there were two St Patricks? One at the top of the mountain and one at the base. What if they started at the same time to descend and ascend along the same path? They would meet each other regardless of how fast each is travelling. When they meet, they are in the same time at the same place.
4. Ladybirds
This problem appears very complex. We can have no way of knowing which way they’re facing or where they are at the start. This problem is so complex indeed, that you might suspect that there’s a way of simplifying it.
If you look at the problem differently, you might see that, for our purposes, two ladybirds
walking up to each other and immediately turning around is the same as two ladybirds walking right by each other.
In that case, the longest it will take is, for one ladybird at one end facing the other end, to walk the full length of the stick. At 1 metre/min that would take 1 minute.
These problems, which seemed very complicated, could be resolved once we try to look at the problem differently.
I’m afraid I can’t offer any sure-fire method to be able to spot these.
Luck does undoubtedly play a part but, to borrow from Gary Player, the more puzzles you work on the luckier you will get. The key to improvement is to reflect on how you approached a puzzle, what strategies worked and what didn’t. What strategies could be useful again?
The answers to today’s puzzles
1. Each tyre travels 16,000 km.
2. They each get 3 full bottles and 2 empty bottles and the they each could
make €31.
3. 60 minutes
4. 45 minutes
Solution 1
Six tyres must share four places over 24,000km
24,000 *4/6 = 16,000
It might not be immediately apparent that this is a valid way to solve the puzzle.
We might try to think about it in the following way. What is happening?
We have four wheels and we change the tyres in such a way that each tyre is on for the same distance.
We might make things a bit easier if we simplify the situation.
It is likely that you would change the tyres in pairs, so we can think of three sets of tyres rather than 6 tyres. In fact, you could consider a two-wheel vehicle (a motor cycle) with three tyres, which is equivalent.
Consider a four-wheeled vehicle with an extra pair of tyres. If you swap a pair of tyres after half way, both those pairs will travel half way (12,000km) while the pair left on will travel the full distance (24,000km). No matter when you make the change, one pair of tyres will stay on longer than the other pairs. That suggests to us that the only way to have all tyres on for the same distance is that the tyres taken off must be put back on again. As we have three pairs, dividing the total distance by 3 might be an appropriate place to start.
After 1/3 of the distance, 8,000km, try changing one pair of tyres.
After another 8,000km one pair has been on for 16,000km. Take that pair off and put back on the pair first taken off. The pair kept on has already travelled 8,000km; the pair put back on has already travelled 8,000km. They travel together the last 8,000km. Therefore, each tyre travels 16,000km while the vehicle travels 24,000 km.
Solution 2
They each get 3 full bottles and 2 empty bottles and they each could make €31.
They can do this by adding the contents of two half-full bottles to two other half-full bottles. That makes two more full bottles and two more empty bottles and leaves one half-full bottle.
They now have 9 full bottles, 1 half-full bottle and 5 empty bottles. The obvious thing seems to be to drink that half bottle between them. That leaves six empty bottles. Altogether, that is now 9 full bottles and 6 empty bottles. They each get 3 full bottles (€10 each) and 2 empty bottles (50c each).
To those who will say that it is unethical, or even unlikely that they could pass off
previously opened bottles to the innkeeper, note that the innkeeper is unscrupulous. It is also a puzzle, so a little liberty can be taken and a bit of lateral thinking is allowed.
Solution 3
If you haven’t seen this sort of problem before or you aren’t familiar with fluid mechanics, you might find it hard to make a start here.
After all, you don’t know the volume of water required to fill the bath.
The filling and emptying of the bath is independent of the volume. However, you could guess a volume just to get started. An overflowing bath probably has over 200 litres of water. But, I would guess 60 litres here to start. This is because 60 is divisible by 12, 15 and 6, and when we see related values in a puzzle we are alerted to the possibility that these values may have been chosen so that fractions will work out cleanly.
Assuming that the volume concerned is 60 litres:
The cold water tap fills 60 L in 12 minutes. The cold water flowrate would be 60/12 = 5 litres per minute.
The hot water tap fills 60 L in 15 minutes. The hot water flowrate would be 60/15 = 4 litres per minute.
The drain will empty the bath in 6 minutes. The out flow will be 60/6 = 10 litres per minute.
The flow into the bath will be 5 L/min (cold tap) plus 4 L/min (hot tap). That is 9 litres flowing into the bath per minute while 10 litres flow out.
Overall, this would be the same as having the taps off and 1 litre flowing out per minute.
At 1 L/min, it will take 60 minutes for 60L to flow out.
If you were to try this again and assume any other starting volume, you would get the same answer.
If you were familiar with fluid mechanics and algebra you might approach it like this:
Volume divided by time is flowrate
Initial volume = V (L)
Flowrate = Q (L/min)
Time = t (min)
Q = V/t
t = V/Q
Net flow, Q, is
Q = V/6 – V/12 – V/15 (as above with V instead of 60)
= V(1/6 – 1/12 – 1/15)
= V (10 -5-4)/60
= V(1/60)
Q = V/60
Remember t = V/Q
t = V/V/60
= 60(V/V) (the Volume cancels out, showing we don’t need to know the volume
for this problem)
= 60 minutes
Solution 4
The swimmer covers 1 kilometre in 45 minutes in still water.
If you thought it was 1 hour, you are not alone.
Intuition would suggest that it would be the average of the two times, 0.5 hours and 1.5 hours. Intuition is very important in problem solving, but we must check our intuition.
In fact, it is the average of the two speeds we should be looking at, rather than the average of the two times.
The speed going with the current is easiest to calculate: speed is distance divided by time.
1 kilometre (km) in 0.5 hours is 2 kilometre per hour (kmph).
1 km in 1.5 hours: (1.5 hr is 3/2 hours) = 1/(3/2) = 2/3 kmph.
This indicates the river current is 2/3 kmph and that the swimmers speed in still
water is 4/3 kmph.
Going upstream the current works against her and her speed is 4/3 – 2/3 = 2/3 kmph
Going downstream the current assists her and her speed is 4/3 + 2/3 = 6/3 = 2 kmph
So finally we calculate how long it would take her in still water (at a speed of 4/3kmph):
speed is distance/time
Therefore:
time = distance/speed
Time = 1/(4/3) = 3/4 of an hour, or 45 minutes.
If you are still incredulous about the answer, consider that the swimmer has been working against the current for an hour and a half and only enjoying the assistance for a half an hour.
Therefore, we might expect the time to be below one hour and above a half an hour.
Keep practicing, keep reflecting, keep thinking!
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