IT’S FRIDAY OF Maths Week… let us know in the comments how you’ve been doing with our puzzles.
The Less Than a Euro Shop
In Maths Week, we like to show different aspects of mathematics perhaps not touched on in school. But basic numeracy is so important for people to participate fully in society.
For instance, in daily shopping, basic maths is needed.
We’ve all been to the Euro Shop (or the €2 Shop), but have you ever been to The Less Than a Euro Shop? Every item they sell costs less than €1.
Perhaps some cost 10c or 24c, others, 59c or 70c, a few might cost 95c or 99c. However, no prices are listed; prices can only be determined by scanning.
Suppose there are two items you particularly wish to know the prices of, packages of frozen apples and tins of bananas.
You ask for help and are told that you can scan any item or combination of items you wish and the giant scanner will give the total cost.
Today, alas, only one scanner is working, and even it has some limitations. The scanner can scan any number of items at once but it will only tell you the total cost of all the items.
Unfortunately you can only use it once.
For instance, scanning 3 packages of frozen apples and 22 tins of bananas might yield the total cost €14.32. But that’s not enough information to work out the individual costs of packages of apApples and tins of bananas.
Separately scanning just one package of apples, and then a single tin of bananas, would of course do the trick. However, the scanner is on the blink, and after a single use it will shut down for hours: using it twice simply isn’t an option.
1. How, with a single scan, can you deduce the individual cost of packages apples and tins
of bananas?
2. Now imagine that you are interested in the unpriced jars of pickled clementines as well. How, with a single scan, can you deduce the cost of all three types of fruit: apples, bananas and clementines? (You might need a credit card with a high upper limit!)
Come back tomorrow for the answers to today’s puzzle.
Thursday’s puzzle: the answer
There are two things that make this demonstration of presidential powers possible. One is mathematical, and the other is magical.
First we’ll explain the maths part.
These are the nine Irish presidents so far: Douglas Hyde, Seán T. O’Kelly, Éamon de Valera, Erskine Childers, Cearbhaill Ó Dálaigh, Patrick Hillery, Mary Robinson, Mary McAleese, Michael D. Higgins.
The president chosen by your friend results in the spelling out of a name in two stages (each time dropping the remaining cards on top as a unit), then a repetition of the spelling out of the first word (and a final dropping of what is left). Each spelling out of part of a name reverses the order of several cards before the rest and are dropped on top without changing their order.
Also examine the nine names: for each president the total number of letters in their name is 11 or more. It’s 11 for Douglas Hyde and Seán T. O’Kelly, 13 for Éamon de Valera, 15 for Erskine Childers, 18 for Cearbhaill Ó Dálaigh, 14 for Patrick Hillery, 12 for both Mary Robinson and Mary McAleese, and 15 for Michael D. Higgins. (Note: Both de Valera and Ó Dálaigh are spelled out as 8-letter last names, and the middle initials of O’Kelly and Higgins are considered as the last letters of their first names.)
Now let’s see what happens when we use these names to deal out some cards, and drop the others on top, three times over.
Take any Ace, 2, 3, …., 10 and Jack from a pack of cards—suits are not important—and arrange them in that order, face down, with the Ace on top and the Jack on the bottom.
Spell out the letters of Robinson, as you count out 8 cards to the table, and drop the rest on top. Pick up the packet and inspect it. You will find that from top to bottom it now runs: 9, 10, Jack, 7, 6, …, 2, Ace.
Turn it face down again and do the Mary spelling and dropping part. When you inspect it this time, it runs 7, 6, …, 2, Ace, 8, Jack 10, 9, from top to bottom. Turn it face down and do the Robinson spelling one final time. The cards now run Jack, 10, 9, 8, Ace, 2, …, 7. Do you notice anything interesting?
Let’s do this all over for Sean T. O’Kelly. Reset the packet so it once again runs Ace, 2, 3, …., 10, Jack from top to bottom. After the first O’Kelly stage we get 7, 8, .., Jack, 6, 5, …, Ace. Then Sean T leads to 6, 5, …, Ace, Jack, 10, …, 7. Finally, the second O’Kelly spelling results in Jack, 10, …, 7, Ace, 2, …, 6. What to you notice this time?
One last experiment: you can check that if we use our current president, Michael D. Higgins, starting with Ace, 2, 3, …., 10, Jack we end up (after the three spellings) with Jack, 10, …, 4, Ace, 2, 3. Isn’t a pattern emerging?
Each time, the top card at the end is the card that started on the bottom, and the one beneath that was originally two from the bottom, and so on. There is a lot of surprising regularity here, and this kind of thing works more generally. If we use a packet of just 9 cards, and we use a name in two parts where the total number of letters is 9 or more, the same kind of thing works.
Since we have to able to accommodate Cearbhaill Ó Dálaigh, that’s not an option for us.
Remembered as we revealed yesterday, you need to know the identity of the bottom card at the outset. After you take back the mixed up 11 cards from your friend, peek at the bottom card as you tap the packet on the table to straighten up the cards. Just make sure nobody sees you doing this.
Maths and magic, a potent mix!
Parting thought: see what happens to the Ace to Jack stack if, after doing the “Robinson, Mary, Robinson” routine, you do a fourth spelling, this time a repeat of “Mary”. Are you surprised? Try it for “O’Kelly, Sean T., O’Kelly, Sean T.” The same lovely surprise should happen.
Maths Week Ireland is coordinated by SETU with partners across the island of Ireland. This year over 400,000, north and south will take part and these puzzles give you a chance to participate.
The Maths Week puzzles this year are presented by Colm Mulcahy, professor emeritus of Mathematics at Spelman College, USA, and adjunct professor with Calmast at South East Technological University. Colm is chairperson of the Martin Gardner Foundation USA, and the curator of the website mathsireland.ie
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