## Mathematical Mindreading: Splitting up Piles

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Here’s another mindreading trick with which to amaze your friends and family! Start with a pile of pennies (or beads or paper clips or poker chips etc.) You’ll need to know in advance how many pennies there are: for this example we’ll assume that there are 20 pennies in the pile. It’s not a big deal if your Audience knows that you know how many pennies there are, although if you do this trick more than once it is most impressive to use a different number of pennies each time.

Ask for a Volunteer from the Audience, and give the Volunteer a calculator if necessary. The Volunteer will take the pile of pennies and, while your back is turned, split it into two piles (with at least one penny in each pile — “empty” piles don’t count). The Volunteer multiplies the number of pennies in each new pile together: for example, if the Volunteer spilts the pile into a pile of 13 pennies and a pile of 7 pennies, the Running Total will be 13·7=91.

Next, while your back is still turned, the Volunteer takes one of the piles, splits it in two, multiply the number of pennies in the two new piles, and add that to the Running Total. For example, if the Volunteer splits the pile of 7 pennies into a pile of 6 and a pile of 1 [so that there are now three piles with 13, 6, and 1 penny] the Volunteer should multiply 6 and 1 to get 6, and add this to the previous Running Total to get a new Running Total of 97 (91+6).

Again, the Volunteer takes one of the piles, splits it in two, multiplies the number of pennies in the two new piles, and adds that to the Running Total. For example, the Volunteer might take the pile of 13 and split it into piles of 3 and 10 [so that there are now four piles with 3, 10, 6, and 1 penny]; the Volunteer will then multiply 3 and 10 to get 30, and add this to the previous Running Total (of 97) to get a new Running Total of 127.

The Volunteer continues to split up the piles and keep track of the Running Total until every pile has just one penny in it. At this point the Running Total becomes the Final Total. Meanwhile, your back is still turned this entire time: you might be reading, or chatting with the audience, or listening to your CD of Dr. Ammondt singing “Blue Suede Shoes” in Sumerian.*

When the Volunteer is all done, you’ll turn around. You can see all the piles of 1 penny each, but of course you have no idea of the way in which the piles were split to get to that point. Ask the Volunteer to think very hard about what the Running Final Total is, and then use your amazing mindreading powers to declare the answer! And the crowd goes wild!

How do you do it? Well, in this case the Final Total will be 190. In general, if you start with n pennies then the total will be n(n-1)/2. This trick can take a while to do, so if your audience is likely to become restless you might want to start with a smaller number of pennies.

There’s a nice proof of the solution here; the basic idea is that you can envision there being strings between all the pennies at the beginning. Since each of the n pennies is connected to the remaining (n-1) pennies, there would be n(n-1)/2 strings (the ½ appearing because each string gets counted twice if you just multiply n by (n-1)). Whenever you split up a pile, think of that as cutting all the strings between the piles: if there are x pennies in one piles and y pennies in the other pile, then you have to cut xy strings between them, and this is exactly the amount that keeps getting added to the Running Total. Once each pile only contains one penny, all of the n(n-1)/2 strings have been cut so that is what the Final Total will be.

*Yes, we own this CD. What’s especially fabulous is reading the liner notes: it includes the Sumerian, but also a translation of the Sumerian back into English (“On my sandals of sky-blue leather do not step!”). And this CD has been positively reviewed by the Archaeological Institute of America! If Sumerian is too obscure, you could always try one of his Elvis tunes in Latin, such as Tedere me ama (Love me Tender) orNunc Distrahor (All Shook Up).**

**Yes, we own this CD as well.