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Maths Week: Friday puzzles

Fancy another mathematics challenge? (And get the answer to yesterday’s puzzles.)

Image: Eamonn Farrell/Photocall Ireland

MATHS WEEK STARTED on Saturday and, as is our annual tradition, we’re setting our readers some puzzles. Give them a go!

Day 7: Change your point of view

We’ve a number of conundrums to keep you busy today…

Often it’s very fruitful to change your point of view. In relationship problems, for instance it’s a good idea to try and see things from the other person’s point of view.

Some problems are only difficult because of the fixed way we look at them.

1. Classic car puzzle

You look out from an office window onto the car park below with the puzzling number system.

What’s the number of the spot where the blue car is parked?

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Tip: Looking at a problem differently might involve physically changing your point of view.

2. Train puzzle 

Two trains are headed towards each other on the same track.

Train A at 40kmph and train B at 60kmph.

When the trains are 100km apart, a fly takes off from the windscreen of train A and flies towards train B. The fly is incredibly fast at 200kmph. It reaches train B, without losing any time, turns around and flies back to train A.

It flies back and forth between the two trains until it is squashed in the collision.

(For some reason, known only to the problem poser, we’re more interested in the progress of a fly than the impending disaster. Thankfully, we can say that the trains were autonomous test vehicles and no humans were injured in the making of this puzzle. Everyone was OK, except the fly!)

So, how far does the fly travel in total?

Tip: Instead of thinking about the back and forth journey of the fly, just consider how long the fly is flying for.

3. Climbing the Croagh

St Patrick arrives at the bottom of Croagh Patrick and at dawn begins his ascent. He stays the night at the summit. At dawn, he begins his descent following the exact same route.

Is there any moment in his journey when he would be at the exact same point at the exact same time on both days? How can we be sure?

4. Ladybirds on a metre stick

Ten ladybirds are placed on a metre stick.

Each walks at 1 metre per minute towards either end of the stick.

If two ladybirds meet, they turn around and walk in the opposite direction. When they reach the end of the stick, they fly off.

What’s the longest time it will take to clear the metre stick?

Come back tomorrow for the answer to today’s puzzles.

Puzzles compiled for The Journal by Eoin Gill of Maths Week Ireland / Waterford Institute of Technology.

Thursday’s puzzles

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The answer to Puzzle One

We know they all lose in turn, therefore it’s Carter who loses the last game.

Carter therefore doubled Adams’ and Baker’s money after the last game.

If that brought them each up to €8, they each had €4 after the second game and Carter must have had €16. Once we have figured this we just repeat the steps.

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Baker lost the second game, so she had to double Carter’s and Adams’ money.

Therefore after the first game, Carter must have been at €8, Adams at €2 and Baker on €14.

Adams lost the first game, so she had to give Baker €7 and Carter €4, so she must have had €13 to start, Baker had €7 and Carter €4.

The answer to Puzzle Two

You may have gained insights from playing the Nim game. Maybe you simplified it and analysed the game with fewer pieces, there are several strategies that are helpful here.

The key is to figure out where you want to be on your second last go.

If you can leave 6 pieces, then your opponent can’t win. If they take 1 piece, you can take 4 leaving the last piece for them. If they take 4, you take 1. If they take 3, you take 2 or if they take 2, you take 3.

So where do you want to be, to be sure to leave 6 pieces? You need to leave 11 pieces on your previous go. If they take 1, you take 4 thus leaving 6.

Where do you want to be after your previous go? You want to leave 16. The points 16, 11 and 6 are like stepping stones.

Because it might be a bit hard to remember them and count them during a game the simple way to remember is: on your first go take 2 leaving 16 and then whatever number your opponent takes, you make it up to 5 (if they take 4 in a move, you take 1, if they take 3 you take 2 etc.).

Of course, if you change the rules of the game, you’ll have to determine a new strategy.

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