Terms:
In this trick, you are the trickster and
the person you are showing this to is the volunteer.
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Preparation:Take
the face cards and 10's out of a standard 52 card deck. Shuffle
these cards and give them to the volunteer, then let the volunteer put
six cards out in a row. An example is shown below:
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The trickster now takes a moment
to prepare and selects a card turned face down above the six cards in a
row at the apex of a triangle. (How to pick the card will be
explained soon.) The table now looks like this:
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Card
Face
Down
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6
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8
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4
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A
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7
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4
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Now you will show the
volunteer how to fill the rows above the first randomly selected row; a
card will be place between each pair of cards that will be the sum of
the two cards. If the sum is more than 9, add the two digits
together and you will get a number between 1 and 9. (Aces count as
1.) Here is what the board will look like when this row is filled
in.
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Card
Face
Down
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6+8 = 14
so 1+4 = 5
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8+4 = 12
so 1+2 = 3
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4+1 = 5
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1+7 = 8
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7+4 = 11
so 1+1 = 2
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6
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8
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4
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A
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7
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4
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Here is the next row
filled in.
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Card
Face
Down
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5+3=8
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5+3=8
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5+8 = 13
so 1+3 = 4
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8+2 = 10
so 1+0 = 1
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5
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3
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5
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8
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2
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6
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8
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4
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A
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7
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4
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Notice that in this case, the number 8 shows up a lot; you may have to
borrow cards from lower row to fill in the rows as the trick progresses. |
Now the row with three
numbers.
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Card
Face
Down
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8+8=16
so 1+6 = 7
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8+4 = 12
so 1+2 = 3
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4+1 = 5
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8
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8
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4
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1
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5
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3
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5
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8
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2
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6
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8
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4
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A
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7
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4
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And now the next to last
row.
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Card
Face
Down
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7+3=10
so 1+0=1
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3+5 = 8
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7
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3
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5
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8
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8
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4
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1
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5
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3
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5
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8
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2
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6
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8
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4
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A
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7
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4
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The trickster now reveals
the card at the top of the triangle, which of course is a 9. How
did you do that, the volunteer will wonder?
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How to pick the card face down: Add
up the 2nd and 5th cards and multiply by 5; then add the rest of the
cards - the 1st, 3rd, 4th, and 6th cards - the running total; this
should be a two (or maybe three) digit number, so add the digits
together. In our example we would get (8+7)x5=75, then adding the
rest we get 75+6+4+1+4 = 90, which results in 9+0 = 9. Again, the
numbers you have to multiply by are as follows:
First card
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Second card
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Third card
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Fourth card
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Fifth card
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Sixth card
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times 1
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times 5
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times 1 |
times 1 |
times 5
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times 1 |
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Why
this trick works: The fifth row of Pascal's Triangle has
six numbers in it: 1, 5, 10, 10, 5, 1. We could multiply the third
and fourth cards by 10, but since we are just going to add digits
together, multiplying by 10 and multiplying by 1 give the same
result. (In math, this is
modular arithmetic we say 10 = 1(mod 9).) Each of the numbers
will be added to the final total 1, 5 or 10 times, and so we can use
this to predict the total before the volunteer gets to it.
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Variation
of the trick: Only
remove the face cards. When a sum is larger than 10,
just use the last digit, so when 8+7=15, the card you would put above
those two cards would be 5. Now, the multiplication to predict the
last card goes uses the numbers below, and the last digit of the sum is
the card you put face down at the top:
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First card
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Second card
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Third card
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Fourth card
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Fifth card
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Sixth card
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times 1
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times 5
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times 0
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times 0
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times 5
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times 1
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