From reader Noble Abraham comes this question:
Is it possible to create a pool of distinct four digit numbers, so that the sum of 8 randomly chosen numbers from this pool is 35712? If yes, how?
I posted this in Quora also.
The basis of this question is actually a performance by a magician. He first wrote down his prediction on a sheet of paper. Then he picks someone from the audience to randomly select 8 four digit numbers from about 1000 (according to him) distinct four digit numbers kept in a bowl. The magician then adds these 8 numbers and finds the sum as 35712, which happens to be his prediction as well. I feel that the pool of numbers has a high probability that sum of any 8 numbers is 35712. Also, think 35712 can be replaced by any 5 digit number, and re-create the pool accordingly.
Am I wrong in asking the question. Or should I think that the fellow has got magical powers. He has chosen at least 11/12 numbers from the bowl that were distinct; 8 for performing the magic and 3/4 to show that they are distinct, before actually going in to the act.
Thank you so much for your valuable time.
You came to the right place because I have studied “Mentalism” i.e. mental magic, i.e. tricks which make it appear that one has paranormal powers, for some time. The top lesson gleaned from years of reading is that the audience never remembers what happened.
I have confirmed this wisdom time and again, both in my own “performances” and in the work of professional magicians. So I hope you won’t take it badly if I suggest you might not have perfectly recalled the exact sequence of events.
Now, there are three broad ways this trick can be accomplished. I won’t tell you exactly how the first two work, but we’ll try to figure out the third. It will be obvious that the methods in all three can be mixed.
The first is substitution. That is, the picking is genuine, as it appears. The magician really does have a bowl of many 4-digit numbers, all different. Eight are pulled out and somebody—probably not him—does the addition. Usually, the magician will have it done on a large chalkboard so that the audience can see the numbers, or he’ll have one or two people use calculators so that the sum can be verified. It helps (but it not necessary) the magician to be able to add rapidly. This is a skill that can be mastered easily.
The sum is announced, perhaps double-checked, then the magician reveals his “prediction”, which is found to match. Off the top of my head, I can name about a dozen ways that the magician “swaps” the prepared prediction with another “prediction” he made after learning the true sum.
The second is a force. Here, the picking is not genuine, despite appearances. There are literally hundreds of ways, with new ones invented monthly, whereby a magician can make it seem that you have a free choice where in reality the outcome is predetermined. (Sort of like how neurologists view all of human behavior.) Of course, he needn’t force every number of the eight picked. He would only have to force enough of them to cause the final solution to belong to a small set (a dozen or less) of possible sums. He could then use any number of methods to reveal the “prediction.” A cheesy, but effective, way to do this is to have, say, four different predictions in your suit pockets, two in the breast and two in the outside pockets: you pull out the one that was the final force. A miracle!
The third is mathematical. That is, the 4-digit numbers on the slips are designed such that picking eight of them force a predetermined single sum of 35712. There cannot be 10,000 slips with all the 4-digit numbers 1000, 1001, 1002,…, 9998, 9999 because, of course, by chance you could end up with the sum 8028 (the lowest possible) or 79964 (the highest possible). So the bowl must be loaded with slips such that the sum is fixed. If such a set of numbers exists, such a trick would be called “self working.” If this method is used, I’d lay my money on a set of sums, not just 35712, married to a substitution.
I saw the solution posted on Quora—the writer there suggests labeling all slips “4464”. This works mathematically, but it makes for a poor performance. You can get away with it only by walking to 8 different, widely separated audience members, have them silently pick a number, then have them add it to the sum shown on, say, a calculator. If you’re blustery enough you might get away with this, but chances are you’ll get caught.
So I leave it for a homework problem for everybody. Does a set of N different 4-digit numbers exist such that pulling 8 out of N leads to a sum of 35712 every time?