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Maths Week: Thursday puzzles

Fancy another mathematics challenge? (And get the answer to Tuesday’s puzzle.)

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

We had a slight technical hitch with Wednesday’s puzzle so we’re giving you a bumper double edition tonight. 

Day 6 Puzzle One

Working backwards is a useful strategy in many cases.

In fact, it can be the only way to solve a problem when we know where we want to end up but aren’t sure how to get there.

Children learn quite young that this is a useful strategy doing mazes – start at the end and work backwards.

The following puzzle is adapted from legendary puzzler Henry Dudeney’s Amusements in Mathematics (1917).

Three people, whose names were Adams, Baker and Carter, played a card game.

They had agreed that whenever a player won a game, that player should double the money of each of the other players — that is, give the players just as much money as they had already.

They played three games, and, strange to say, each won a game in turn, in the order in which their names are given above. But a more curious coincidence is this — that when they had finished play, each of the three had exactly the same amount — €8.

The puzzle is to find out how much money each had when they sat down to play.

In this puzzle, we know the end state but we don’t know anything about the starting conditions.

Therefore, we’ll have to work backwards. Another useful strategy is to draw a table.

We have the conditions after game 3 – what must the conditions be after game 2 etc?

Give it a try and see how you get on. 

Day 6 Puzzle Two 

In problem solving, it’s often a useful strategy to examine the penultimate step.

This might be the last step before the solution, or commonly the last step before you realise you have gone wrong. If you realise you’ve gone astray it might be a good idea to back up and try something different.

We’re going to play a game of Nim, which is a mathematical strategy game where players take away pieces from a gathering of objects. 

The form we want you to play now is for two players and uses 18 pieces laid out on a table (matchsticks, counters, coins, sweets, can be used).

A move consists of removing between 1 and 4 pieces. Players take their moves in turn and the player left with the very last piece on the table loses.

There’s a strategy where you can always win if you go first. Working backward is the key here.

Can you use your problem solving skills to figure out the winning strategy? How many pieces should you leave on your second last go to be sure of winning?

In this way can you identify all the intermediate places you should aim for at each go. It might be useful to use the ‘simplify’ strategy – start with a lot less than 18 pieces and build up. If you can’t find pieces to use, drawing a picture will be useful.

For further exercise, you could change the rules of the game – start with a different number of pieces, allow a different range of pieces to be taken away, have the person who takes the last piece become the winner. Determine the winning strategy in your variation.

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.

Tuesday’s puzzle: The answer 

1) 28 matches are needed for the league of 8 players.

How we got that answer: Simplifying the problem is a useful strategy here. One player will require no matches. Two players will require one match. If another player joins them (3) that match will still go ahead, but each player will play the new player making 3 matches.

Adding another player will still have those 3 matches but each player will play the new player – 6 matches in total. So adding a new player adds the previous number of players to the previous total of matches.

2) 4,950 matches would be needed with 100 teams

You might further notice from the above exploration that for 3 players there is 2 + 1 = 3 matches

For 4 players there is 3+2+1 = 6 matches

For 5 players there is 4 + 3 + 2 +1 =10 matches

So for 100 matches there would be 99+98+97….+1 matches.

There is a quick way of adding these by pairing up numbers -
(99+1) + (98+2) + (97+3) ……(51+49) + 50 (there is nothing to pair with 50)

That is 49×100 + 50 = 4,950

A more sophisticated and versatile approach would be to derive a formula from your observations. The formula n(n-1)/2 (where n is the number of players involved) allows us to calculate the number of matches.

If n=3 we need 3(3-1)/2 = 3 matches (as above).

If n=100 we need 100(100-9)/2 = 4,950 (as above)

The formula can be applied to other situations, such as a number, n, of people that all shake hands with each other. Or the number of teams, n, in a football league (if teams play home and away, you multiply the answer by 2).

If we wanted to understand how people are connected together on the internet or social media, we could use a similar approach. As we model complex real world applications, we need to derive more complex formula. We can programme computers to run these complex formulae on masses of data to give confident predictions of the future. That is the power of maths.

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    Mute James Pelow
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    Oct 25th 2021, 12:14 AM

    Can we please stop propagating the lies of the English media? Brexit did the damage, not the protocol.

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    Mute Vonvonic
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    Oct 25th 2021, 6:42 AM

    @James Pelow: Very well said. They’re actually using it as a distraction. And it’s drawing us into something that has nothing to do with us. Brexit is the problem. End of story.

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    Mute Colm A. Corcoran
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    Oct 25th 2021, 7:00 AM

    You can’t hold a poll asking people if they think the Protocol is good for Northern Ireland without clarifying what the alternative is.

    That’s like asking a child if they think the settlement that their parents agreed to after divorce is good.

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    Mute Oisín Dunne
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    Oct 25th 2021, 8:40 AM

    Let’s be clear… article 16 does not end the protocol. It can suspend a part of it for a short period of time. When the UK says it will trigger A16, call it out for what they want to do….they want to scrap it and force a border on the island of Ireland or a border between Ireland and EU. That’s their plan and I believe it’s been the plan all along. This mess has been made by the UK and the protocol is a plaster. The GB companies that send those goods that will never end up back in the EU (including ROI) need to be better catered for. The issue is that there is no trust between the UK and EU as, so far, the UK hasn’t implemented main parts of the protocol so all at risk goods must be considered guilty until proven innocent.

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    Mute Stephen Campbell
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    Oct 25th 2021, 10:08 AM

    Ok theJournal…. Time to correct your headlines… “Is Brexit bad or good for firms in Northern Ireland?”

    The protocol is a workaround to the main issue, Brexit..

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    Mute Gerard
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    Oct 25th 2021, 8:54 AM

    While I’ve no doubt it has caused some legitimate disruption for businesses heavily linked to GB, how did the study take into account costs (for consultancy etc) that would’ve been incurred without the procotol because they also trade with the EU?

    Or how did it take into account all the paperwork NI businesses save because they can trade with the Republic and the rest of the EU freely?

    All these analyses seem to assume that trade with Ireland was either insignificant, or its continuity was a a given (neither of these are true) and that any disruption with GB is a cost without any quantifiable benefit (again not true).

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    Mute John Vectravi
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    Oct 25th 2021, 10:50 AM

    It’s not the protocol that’s not working. It’s brexit that’s not working.

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    Mute lelookcoco
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    Oct 25th 2021, 11:09 AM

    How dare the EU break away from the United Kingdom. They’ve made things very difficult for everyone, especially the Brits!

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    Mute John Sullivan
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    Oct 25th 2021, 3:22 PM

    By leaving the CU and SM and going for a Sharia Brexit GB turned itself into a legal and regulatory Kaliningrad. Their call-their choice…If they hadn’t CHOSEN that there would be no protocol. They want a hard border in IRL or IRL out the the EU-they will get neither but what they will get is humiliation.

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    Mute andrew
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    Oct 25th 2021, 10:38 PM

    It is improving trade between north and south.

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