# Mathemagic - Magic with coins

By Sam Wheeler Friday, September 12, 2014

Hello and welcome to another edition of Mathemagic! If you joined us last week you may have been trying to solve the puzzle and some of you may have solved it yourselves, check out the first trick below for the solution to the puzzle.

## Coin sense

Grab some coins and try these tricks out on yourself. A handful of coins are mixed and then with your eyes closed you can separate them into two piles with the same number of heads up coins in each pile.

1) Place a pile of coins on the table, you can borrow these or use your own.

2) Ask your friend to shuffle them by picking up all of the coins and shaking them in their hands before placing them all on the table again.

3) Openly count out loud how many coins are heads up (if you can do this secretly then that is even better). Remember this number for later.

4) Cover your eyes and turn away so you can’t see the coins and ask your friend to mix them on the table but without turning any over.

5) Now tell your friend that you are going to separate the coins into two piles and each pile will contain the same number of heads. To make it impressive you are going to do it with your eyes covered.

6) Keep your eyes covered and your head turned away from the coins.

7) Now reach over and one at a time take as many coins as the number you remembered in step 3, however ensure that you turn each coin over to the opposite side and place them all down into a separate pile.

8) When you are done uncover your eyes and take a look at the two piles. There may be a different number of coins in each pile but there will always be the same number of heads showing in each pile. So if one pile has four heads then the other pile will have four heads.

## How does this work?

If, for example you have ten coins and five are heads up, and you happened to remove all five heads coins and turn them over then there would be zero heads in each pile. On the other hand if you removed all five of the tails coins and turned those over then there would be five heads coins in each pile. If you remove only two heads (leaving three behind) then you must also have taken three tails which when you flip them over will become heads to match the three you left behind and the two heads you took will become tails.

In the diagram below heads is signified by white coins and tails is signified by black coins:

So whichever heads you do not take will be left in the original pile, the number of tails coins you take will be equal to the number of heads left in the original pile. When you turn the coins in your pile over then the numbers of heads in each pile will match.

## Heads or Tails

Have a go at this next one. Your friend will get to mix up the coins and then cover any one of their choice. You will be able to determine whether their coin is heads or tails up.

1) Place a pile of coins on the table, you can borrow these or use your own.

2) Ask your friend to shuffle them by picking up all of the coins and shaking them in their hands before placing them all on the table again.

3) Secretly count how many coins are heads up. All you need to remember is whether it is an odd number of heads or an even number.

4) Turn your back and invite your friend to turn over any one of the coins.

5) Then ask them to turn over another coin, and another and another for as long as they want but they must tell you every time they turn a coin over.

6) Every time they turn over a coin you need to switch between thinking “odd” to thinking “even” and from “even” to “odd”.

7) When your friend no longer wants to turn over any more coins you must remember whether you are thinking “odd” or “even” and ask them to cover one of the coins with their hand.

8) Now you can turn back to face them.

9) Secretly count the number of heads. Check to see if the number of heads matches what you were thinking of in step 7 (odd or even). If it matches then your friend is covering a coin with the tails side up. However if it doesn’t match then they must be covering a coin with the heads side up.

## How does this work?

If your friend turns a heads up coin over then there will be one less head and if they turn a tails coin over then there will be one more head. So each time they flip a coin over the number of heads up coins will switch between odd and even. This means that as long as you know if there is an odd or even number of heads at the start then you can track if there is an odd or even number at the end.

If there is an even number of heads at the end and one of them is covered then when you check the coins it won’t match what it should be, so you know the missing head is under their hand. If the count at the end does match what it should be then you know that they must have covered a tails up coin. The same is true if the number of heads at the end should be odd.

## Combining the tricks

If you want to do both of these tricks together then you ideally want to leave the best trick until last. If you do ‘Coin Sense’ first then you are guaranteed to end with an even number of coins heads up (the number of coins in one pile will be equal to the number of coins in the other pile and any number multiplied by 2 will result in an even number), so you can immediately start ‘Heads or Tails’ without needing to count the heads up coins – you already know there are an even number so you can jump straight to step 4 of the ‘Heads or Tails’!

## Coin divination

Do you remember the Three Object Divination trick from the first issue of Mathemagic? You can do an even more impressive (and easier) version of this trick by using three identical bottle caps and a coin:

1) First you need to mark a bottle cap, if you can notice a flaw in one of the caps then use that otherwise you can scratch the top slightly with a key. Make this scratch so faint that nobody but you will notice it.

2) Place the bottle caps in a row on the table and make sure the marked cap is in position one.

3) Turn your back and ask your friend to hide a coin under any one of the caps and to switch the remaining two caps.

4) Now you can turn back to face them and look at the caps. If your marked cap is still in position one then you know that they hid the coin under that cap. If it isn’t in position one then the coin won’t be under the cap currently in position one nor will it be under the marked cap – it must be under the only remaining cap.

5) Now that you know where the coin is hidden your friend can switch the caps as much as they like and you just need to follow where the coin ends up.

## Puzzle

Finally I’d like to end with a puzzle, the answer will be posted next week in the next edition of Mathemagic.

Take a deck of cards and shuffle it thoroughly. Can you deal the deck into two piles but ensuring that the number of red cards in one pile is equal to the number of black cards in the other pile? You must do this **without** looking at the faces of the cards.