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Coumeenoole on the Dingle Peninsula Eamonn Farrell

Maths Week: Your Tuesday puzzle

Fancy a mathematics challenge?

IT’S DAY 4 of Maths Week, as is our annual tradition, we’re setting our readers some puzzles. Give them a go!

TODAY, WE ASK some simple sounding questions about a hypothetical place where there are two types of coin – and no bank notes.

Imagine if there were only €3 and €5 coins available: it turns out that any whole number of euros from €8 onwards (say €17 or €91) could be made up using only these coins, though many coins might be involved. [Some lower totals are impossible to achieve, namely, €1, €2, €4 and €7.]

The same cannot be said if only €3 and €6 coins are available; can you see why?

Let’s transport ourselves to the Kingdom of Kerry, and pretend that the only currency there is the mighty dingle.

We ask some questions that can be answered with a little effort, and end with a similar question whose answer is surprisingly elusive.

First, supposed that there are only 5 and 7 dingles coins, nothing else.

We can’t pay exactly for items costing under 5 dingles, or for things costing 6 (= 3 + 3), 8 (= 3 + 5), 9 (= 3 + 3 +3) or 10 (= 5 + 5) dingles in total. Items costing 11 (or 13) dingles also cannot be paid for exactly. Can you see why?

It turns out that things costing 100 dingles or more can always be paid for exactly.

So the crossover from “may not be possible” to “is always possible” is somewhere between 13 and 100. Can you find it?

What is the smallest positive whole number so that for that amount or any larger whole number amount, it can be made up using 5 and 7 coins?

Let’s call this the tipping number for 5 and 7. Would something like this have worked if we had used only 2 and 6 dingle coins? Or 6 and 15 dingle coins?

Now repeat using several other possible pairs of values, such as 4 and 9, or 6 and 7. Namely, find the tipping numbers for those pairs of values.

Do you see a pattern? Is there a general formula that gives the tipping number no matter what two coin values we start with?

If there were, then you could speedily find the tipping number for 8 and 11 without doing any hard work!

Finally, try the same kind of question assuming there are three coin values available, say, 5, 7 and 11 dingle coins. Repeat using three other coin values? Do you notice a pattern here?

Is there a formula that gives the tipping number for any three coin values?

Come back tomorrow for the answers to today’s puzzle.

Monday’s puzzle: the answer

A. Sinead should give Imelda €85.

Sinead pays for the train (€86) and the concert tickets  (€120), totalling €206. Imelda pays €330 for accommodation and €46 for drinks. A total of €376.

Imelda paid €170 more than Sinead. If Sinead pays half of this to Imelda, they will both have contributed €291.

B. We agreed that Imelda spends more than Sinéad, so divide that excess expense into two equal amounts. If Sinéad pays one of those (a half of the discrepancy) to Imelda, all is well. Now each person has spent the same amount.

C. Suppose for the sake of argument that Tom, Dick and Harry respectively spend €320, €300, and €250. Using the method from before, if Dick first gives €10 to Tom, then Tom, Dick and Harry respectively are out of pocket €310, €310, and €250. So Harry is contributing €60 less than either Tom or Dick. If he gives €20 (a third of the discrepancy) to each of his two pals, the overall contributions of the three lads will finally be equal, namely, €290 each. There are other ways to arrive at the same result, but the upshot is the same.

D. It should be clear how the ideas above can be extended to four (or more) people faced with the same challenge of evening up expenses. It is helpful to list the people in order from biggest to smallest spender before doing the calculations.

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    Mute Sean Ryan
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    Apr 2nd 2020, 8:37 AM

    Fair play. She genuinely seems to be a caring person. Love her work for literacy too.

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    Mute Connoroconner
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    Apr 2nd 2020, 9:17 AM

    Don’t laugh now, but Dolly Parton is a true hero of our times. She is a remarkable woman at every level and has done huge amounts of charity work, invested millions in the impoverished local community she came from, has been influential in women’s rights and gsy rights and has written at least 2 all time classic songs.

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    Mute Aisling
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    Apr 2nd 2020, 10:23 AM

    @Connoroconner: it’s true! If you dig below the surface you quickly realise how remarkable she is.

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    Mute Connoroconner
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    Apr 2nd 2020, 12:18 PM

    @Aisling: there was a Sky tv show ‘Brian Johnstons Life on the Road’ which featured an extended interview with her. Brian, front man for AC/DC, and i think he was in love by the end of the show. It captures what a warm, beautiful person she is and shows what a great sense of humour she has too. Worth a watch if you can track it down.

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    Mute Barry Vickers
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    Apr 2nd 2020, 4:10 PM

    @Connoroconner: I’m only going to disagree with you about the last part. Its way more than 2 all time classics. Huge Dolly fan here.

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    Mute Connoroconner
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    Apr 2nd 2020, 9:43 PM

    @Barry Vickers: well i also thinks it’s more, but for the general non country music fan, there are 2 they would know. That’s why I said at least 2!

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    Mute MollyMsalone
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    Apr 2nd 2020, 8:36 AM

    Any sign of Bonio or Bob?

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    Mute chinaski
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    Apr 2nd 2020, 8:40 AM

    @MollyMsalone: Bono actually added to people’s misery by signing a song for them.

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    Mute Tordel Back
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    Apr 2nd 2020, 8:48 AM

    @MollyMsalone: Give Bob a break. He’s made mistakes, but he’s a righteous dude.

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    Mute Damien
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    Apr 2nd 2020, 9:02 AM

    @Tordel Back: Hes a dose.

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    Mute Joe Toner
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    Apr 2nd 2020, 9:26 AM

    @MollyMsalone: leave him out of this…. We are suffering enough without listening to him!

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    Mute Conall
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    Apr 2nd 2020, 10:44 AM

    @Damien: To be fair, he’s both a righteous dude and a dose simultaneously. First bit’s more important.

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    Mute Gerry Mc Carthy
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    Apr 2nd 2020, 11:25 AM

    @Tordel Back: “self righteous “ dude you mean . Dolly Parton can sing intelligible songs and donate substantial money to help people . Bono/Geldoff do neither, but in their imagination think they were important to Ireland’s “ progress “ , God help us !

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    Mute School4work
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    Apr 2nd 2020, 10:00 PM

    @MollyMsalone:

    Bonzo and Boob more like.

    I heard that Paddy Power is donating 1 Billion from the gambling profits:

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    Mute Tordel Back
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    Apr 2nd 2020, 8:46 AM

    The world has never deserved Dolly Parton. A truly decent person.

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    Mute Jake Kelly
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    Apr 2nd 2020, 8:31 AM

    Working 9 to 5 for a way to make a vaccine

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    Mute Karllye kripton
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    Apr 2nd 2020, 9:15 AM

    Thank you dolly, can u imagine if every millionaire and billionaires donated a mill each all over the world

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    Mute New Remedy Europe
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    Apr 2nd 2020, 9:42 AM

    Closer to home a huge thank you to a Conor MCGregor for his 1m donation to aid Irish hospitals in this crisis.

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    Mute Aaron Luke
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    Apr 2nd 2020, 10:12 AM

    @New Remedy Europe: spot on . A lot of people knock connor . But he is a top man . Didn’t he give some kid 10 k who needed an operation recently ?

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    Mute Tricia G
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    Apr 2nd 2020, 9:26 AM

    The world doesn’t deserve Dolly Parton!

    Between her literacy program, giving financial assistance to those impacted by the Tennessee fires and now this (as well the other work through her foundation) the woman is a bonafide saint!

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    Mute Declan Cosgrave
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    Apr 2nd 2020, 9:42 AM

    Wheres Bono , Geldof , JP McManus , Magnier , D.O.B ?????
    Holed up in their caves counting their money

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    Mute Jono
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    Apr 2nd 2020, 10:01 AM

    @Declan Cosgrave: are they doctors ?? Money is not an issue for governments so unless they are doctors we dont need anything from them

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    Mute Will
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    Apr 2nd 2020, 10:02 AM

    @Declan Cosgrave: McManus and Magnier have donated something like €80 million to charity in the last two years.

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    Mute Will
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    Apr 2nd 2020, 10:04 AM

    @Jono: “Money is not an issue for governments”

    With a national debt of well over €200 billion, money is most certainly an issue for our government and for all of us who ultimately must pay such a colossal bill.

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    Mute Connoroconner
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    Apr 2nd 2020, 12:20 PM

    @Will: yeah if you class horse racing and GAA as charities. I’d prefer if they’d pay their taxes here.

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    Mute Will
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    Apr 2nd 2020, 1:21 PM

    @Connoroconner: McManus has a charitable fund he set up years ago to help with all sorts of good causes. There’s a website that goes into the details of it.
    The money is used for healthcare, education, sport etc. Lots of small charities in McManus’s part of the country depend on it.
    Fair enough, be nice if he paid his taxes here but ask yourself, if you, as a billionaire, could half your tax bill by moving residence would you still stand on principle?

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    Mute Connoroconner
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    Apr 2nd 2020, 2:06 PM

    @Will: yes I like to think I’d consider it the patriotic thing to do. How many billions does one person need? Michael O Leary and Bono, despite the abuse he gets, continue to pay their personal income taxes here when they could easily choose not to.

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    Mute School4work
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    Apr 2nd 2020, 10:03 PM

    @Declan Cosgrave:

    And Paddy Power too.

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    Mute
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    Apr 2nd 2020, 9:20 AM

    It’s good to see reports of millionaires ACTUALLY donating. Too often I see the journal blowing up the headline for a “pledge” or a “promise” or a “plan” than never happens. Like the Notre Dam fire, loads came out to pledge money, few really coughed up.

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    Mute Jono
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    Apr 2nd 2020, 9:54 AM

    Not everyone wants to publicly announce their donations..food for thought

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    Mute Paul O Connor
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    Apr 2nd 2020, 9:21 AM

    Good on her for the gesture but realistically money won’t solve this. Medics already have unlimited access to funds. Unfortunately it’s gonna take a lot of time.

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    Mute Paul Atreides
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    Apr 2nd 2020, 1:57 PM

    What more good things can we say about Dolly Parton.

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    Mute John Tierney
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    Apr 2nd 2020, 9:55 AM

    Bill Gates has enough millions. He doesn’t need Dolly’s.

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    Mute Brendan Deegan
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    Apr 2nd 2020, 12:05 PM

    BOB AND BONO are organising a fund raiser.

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    Mute BOH®
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    Apr 2nd 2020, 3:29 PM

    And people say she’s just a big pair of t-ts.

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    Mute Vlad Eire
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    Apr 2nd 2020, 12:33 PM

    Working 9 to 5 (from home)

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    Mute colin weafer
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    Apr 2nd 2020, 3:40 PM

    she has Hugh assets

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    Mute BM
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    Apr 2nd 2020, 7:04 PM

    And people say shes just a big pair of tits

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    Mute roberto Mankini
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    Apr 2nd 2020, 10:51 PM

    Are her big pair of tits to named bob and bono?

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