mind probe

Mind ProbeAuthor(s): Martin HansenSource: Mathematics in School, Vol. 21, No. 1 (Jan., 1992), pp. 2-6Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216423 .

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A popular children's conjuring trick, which I am confident that all readers will have stumbled across at one time or another, involves the set of six cards which are reproduced below in Figure 1.

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.... . . ....

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. . . ................................................... .................................................

........ ............ . . ......

1 3 57 2 3 67 4 5 67 9 11 13 15 10 11 14 15 12 13 14 15

17 19 21 23 1819 2223 20 21 22 23

25 27 29 31 26 27 30 31 28 29 30 31 33 35 37 391 34 35 38 39 36 37 38 41 43 45 47 42 43 46 47 44 45 46 47

49 51 53 55 5151 54 55 52 53 54 55

57 59 61 63 581 59 62 63 60 61 62 63 .. .. . . . .. . .. . ... . .. . . . . .. . . . . . . . . .. . . . . . . .. . . ... . . .. .... .. .. .. . .. . .. . ...*.. ... .. .. . ... .. .. ... .. .. .. .. . .

8 9 10 11 16 17 18 19 32 33 34 35 12 13 14 15 20 21 22 23 36 37 38 39 24 25 26 27 24 25 26 27 40 41 42 43

28 29 30 31 28 29 30 31 44 45 46 47 40 41 42 43 48 49 50 51 48 49 50 51

44 45 46 47 52 53 54 55 52 53 54 55 56 57 58 59 56 57 58 59 56 57158 59 60 61 62 63 60 61 62 60 61 62 63

= . -.... ...-.. .. ...- . ....- .. ..-.= ...- . ..- . ... .w .. .... .. ...= .... . = . .. = = = ..=.. .= .. . . .

-.... . . . . . =. . -. -.. -... -... ..=. .. .

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The performance of the trick involves the magician asking that a number between 1 and 63 (inclusive) be memorized. He then presents each card in turn asking each time if the number being thought of is on that card. Having done this six times the Mind-Probe is complete and the magician announces what the thought of number is.

The secret, just in case some readers don't know it, is to add together the first number on all cards that received an answer of "Yes".

Understanding and constructing these cards is an activity that I have found my classes enjoy. Furthermore, it is the doorway into an investigation which can encompass the mathematics of number-bases, logic, and the building of mechanical, cardboard computers! Ah yes, computers; my cue to begin by reminding readers that modern, digital computers work in binary.

The first step in constructing The Simple Set of Mind-Probe Cards is thus to convert our everyday denary, or base ten numbers, into the ones and zeros of binary and this is best done by means of a table as shown in the first half of Figure 2. The full table is of seven columns by sixty-five rows so either some sellotape or some long paper is needed! The ones and zeros are to be written lightly and small, for the zeros are not wanted, and the ones are replaced with the number that gave rise to their location in the table. This will become clear upon glancing at the second half of Figure 2 which shows the finished result.

The cards are obtained by simply cutting up the table into six very long strips and disposing of the table column headings and left hand row numbering of 1-63.

Having grasped the principles of the binary Mind- Probe it occurred to me that one could concoct a similar

2 Mathematics in School, January 1992



No 2s ...23 22 f 20 NO 2s . 23 22

. 20

3 0 0 0 0

1 3 0 0 0 0

1 1O

4 0 0 0 1 0 0

4 0 0 0 1 0 0. 5 0 0 0 1 5 0 0 0

..... 2 618 4 161 4 2 1 iiiij

5 4 2 0 4 3 2 1

.....

but more advanced set of Mind-Probe cards using the mathematics of base three numbers.

This results in fewer cards being needed to cover a given number range but the questioning process is now more complex. For example, to be able to say "Think of a number between 1 and 80" needs only the four cards shown in Figure 3 but one now not only has to ask for each card, "Is the number being thought of on this card?"

I 4 7 3 4 5

-19 I 22 25 1211 221 23 p

28s 31 34 _30 31/ 32 3 7 40 43 39441

It46 49 I~ 52 48 491 50 . . . . . . 6 1 . 8 9 I.. . . . 64 * 670 70 -66 67 68 *I 73 76

..7.. 76 77.

9 10 11 12 27 281 291 30 31 32

15-ls 161 17llt~~l- il33 34 35 36 3738 : 4:1 i 391 40 41 42 431 44 j

_361 37 381 39 40 411 :45 46 47 48 491 50 :1 421 433 44 42

63 64163 66

69 702 71 22

Vig.

but if the answer be "Yes", follow up with "Is it written in black or white?" The secret of evaluating the thought of number this time is to add together the first number of the stated colour on "Yes" cards. Figure 4 illustrates the construction process of The Base Three Mind-Probe.

Clearly, one could pursue this process further to develop other sets of cards that used other number bases for their construction and other questioning procedures to enable the thought of number to be found. However, a more exciting development is to give the cards a little mechanical "intelligence" so that the cards themselves "work out" the number in mind. This is achieved by means of cutting holes, or windows, in the cards and placing them into a pile in such a manner that the thought number only, shows

~saa la1 2 131

~1 11 7 l.................~

27 9 3 7 9 1 3 3 3 3 No 3 3 3 3

14 0 1 1 2

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through the window. For our earlier binary Mind-Probe the cards are made into squares, each with a square four by four grid centrally placed upon it as shown in Figure 5. The fourth card is double sided and it is important that the back be correctly orientated with respect to the front. I recommend duplicating this card as shown and then folding along the broken line and gluing the two halves together "back to back" to form the single, double sided card.

Once again, our Mind-Probe begins with the request "Think of a Number". This time, it is to be "between 1 and 15". Again each card is shown, in turn, and it is queried whether or not the number being thought of is on the card. If the answer is "Yes", put the card down so that the "Yes" printed upon it is on top. If the answer is "No", turn the card so that the "No" will be on top. Stack the cards one over the other, with the double sided card that has no hole in it being put down last, over the others. Pick up the cards and turn the stack over. The correct number will show through a window in the cards.

Of course, numbers are not the only items that can be placed upon the cards. I have a set upon which I placed the twelve signs of the Zodiac plus a "DN" for "Don't

Mathematics in School, January 1992 3



Know" and a "C" for "Cusp" and these can be used to work out people's astrological birth signs. I have another set featuring "drinks" which can be used as a "I'll guess what drink you would like" mathematical trick at parties. If it is desired to place more than 16 items upon the cards then the number range can be extended to 1-63 by constructing the larger cards of Figure 6. In this diagram, I have shown only where to cut the holes as the numbers to be placed upon the cards are those of our very first set of Mind-Probe cards. I leave it to enthusiastic readers and their pupils to work out what goes where on the back of the sixth card, which can be easily done by performance of the trick for each of the 63 numbers in turn, writing the appropriate number through the window and onto the back of the sixth card each time.

A further logical development is to, once again, move on into a higher number base in much the same way as

all letters so there is no need to ask if the thought letter is upon it. It is placed last in a face down manner on top of the stack prior to turning over and revealing the letter in mind.

The logic upon which these cards work can be isolated and used to produce sets of cards for solving logic problems. One such set is given in Figure 8 and is my development of a system originally devised by the ubiquitous Martin Gardner of The Scientific American. I suggest that you photocopy Figure 8 and then carefully cut out the fourteen cards and their windows which are the black areas upon them. A razor sharp knife and metal ruler will ensure a smart set is cut.

Their use is best illustrated by an example, so consider the following problem which stars two girls, Ann (a) and Betty (b), and a boy, Chris (c). They are thinking about joining a "Maths Club" at their school. However, although

The Mind.Probe "Computer" Cards.

Back of Top Card

Figure 5

was done with the earlier "non-computerized" cards. Attempting to do so is not straight forward, and I suspect that my efforts here could be bettered by a whizz kid, but I have managed to construct a set of Alphabetical Mind- Probe Cards using base three numbers. These I proudly present in Figure 7. As with the previous base three cards the questioning is a little more complex and one must ask first if the thought of letter is upon the card and follow up "Yes" answers with a colour query. The positioning of the cards in the stack is face up and such that the word "Cards" is along the base if a "No" answer is given, "Mind" is along the base if a "Yes and it's coloured white" answer is given, and "Probe" is along the base if a "Yes and it's coloured black" answer is given. The fourth card contains

(at their teacher's insistance) one of the girls must join the club, if Ann joins then Betty (who dislikes her) will not join. Furthermore, Chris will only join if Betty (his girl- friend) does. What combinations of the three pupils might join the club? With a little thought, the answers are clear. Either Ann joins on her own, or Betty joins alone, or Betty and Chris join together. The interest is, of course, in how to achieve these answers from the cards.

First the base card is placed down such that the words "Base Card" face the reader. The wording from the question that "One of the girls must join the club" causes selection of the OR card and this is placed on top of the base card (i.e. ANDed with it) such that the words "a or b" are along the base.




A 1.63 Mind-Probe "Computer".

Figure 6

ASet Of Alphabetical Mind-Probe Cards.

Cut out triangles shaded thus; Figure 7

Mathematics in School, January 1992 5



A Set of Logic Cards for solving Logic Problems.

Figure8

"If Ann joins then Betty will not" converts to "if a then NOT b" and so the card "if a then "b" is added to the pile.

"Chris will only join if Betty does" is slightly more awkward but (if you think about it!) means that "If Chris joins then Betty will" which is "if c then b" and so that card is also added to the pile. Showing through the windows from the base card are the expected solutions. "a, ^b, "c" which is Ann on her own, '"a, b, "c" which is Betty on her own, and "a, b, c" which is Betty and Chris together.

The fascination of this final set of cards is as much in inventing problems that can be solved using them as in the actual solutions themselves. I am sure that enthusiastic readers will have no difficulty in taking things further by devising some ingenious problems of their own.

Reference (Logic Cards) Cundy, H. M. and Rollett, A. R. (1961) Mathematical Models, Tarquin

Publications.




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