Lucky 7 - A Card Trick
A mathematical trick is one whose success doesn't depend on any conjuring skills. I described such a trick in PLUS magazine last Autumn. Stephen Hard exacted recenge for two years of Pure Mathematics by showing me another as I was dashing off to my lesson this Friday Lunchtime and insisting that I prove why it always works.
Firstly, he asked me for a number between 1 and 19 - I chose 8. From a shuffled face down pack I was to discard the top 8 cards. Then I dealt a pile of 26 cards face up which were then turned over and placed beneath the remaining cards. The top 3 of this pile were turned over and placed in a row - they happened to be a 2, a 4 and a King, which, I was told, gave me a score of 16, as all face cards were to count 10. Next, I was to "make the cards up to 10" by dealing 8 cards on the 2 and 6 on the 4 (with no cards required on the King). Since my score was 16, Stephen told to choose the 16th card dwon from the pile remaingin and confidently asserted that it would bte... the Queen of Hearts! It was. I was impressed. I couldn't see how all the steps that I had performed could fit together in a way which guaranteed success. Eventually though I managed to sort out what was going on. Have a go yourself. You'll find some simple algebra comes in handy.
To see why the trick always works we'll first show that, despite the elaborate procedures used, the card which eventually chosen is alwasy the 33rd down in the pile of cards which remain after some have been discarded. Suppose that the 3 cards to be made up to 10 have values x, y and z. On these will be dealt a total of (10 - x) + (10 - y) + (10 - z) cards, which, with the three cards themselves, make 33 - (x + y + z) cards. The x + y + zth of those remaining must therefore always be the 33rd. Now we need a way for the magician to identify which card will occupy this position. If the number of cards initially discraded is d, s/he can do this by noting the d + 7th (hence Lucky 7) from the bottom of these 26 face up cards and hence the d+7th down when they are turned over. When these are placed under the 26-d cards remaining, it will be the 26-d+d+7th down, i.e. the 33rd.
I like this trick, although I feel that it would benefit for the addition of a suitable storyline to accompany the various actions; you know, something along th elines of "Within this pack is an impostor, which through my magic powers I shall expose…". And Lucky 33 might be a more appropriate name!