Texas Instruments International Conference Chicago, IL 2012
[email protected] Adapted from NCTM Mathematics Teacher,
March 2011, "AN EXCEL-LENT CARD TRICK" by Holly S. Zullo. Slide 2
Card Trick Use any 15 cards from a standard 52 card deck of playing
cards. Ask a volunteer to select one card, note the suit and
denomination, and then put the card back face-down in the deck.
Shuffle the 15 cards and then layout the cards face up, one at a
time, to form 3 piles. Slide 3 First card goes into pile 1, the
second into pile 2, and the third into pile 3, and the fourth card
on top of pile 1 and so on. Ask volunteer to identify the pile
their card is in; the pile is one of the columns. Lay out the cards
as shown, this layout is Layout 0 : Slide 4 Ask the volunteer to
identify the pile containing their card. (this is the key pile; the
card will be the key card) Pick up the piles one at a time, making
sure to pick up the key pile second. As you pick up each pile, keep
the cards face-up and put the next pile beneath the first pile
picked up. Pick up and Reassemble Slide 5 When finished picking up
piles, turn over the stack so that the cards are face down.
Beginning with the card on top of the deck, lay out the cards in
the same fashion, this will be Layout 1. Ask the volunteer to
identify the pile containing the key card again, and pick up the
piles with the key pile in the middle again and then layout the
cards again. This will be Layout 2. Repeat the process three times.
Slide 6 Reveal the Key Card! Ask the person to identify the key
card pile once again and pick up the cards in the same fashion. The
key card now be the eighth card in the deck. Creatively layout the
cards facedown or count out the cards. Keep track of the eighth
card and then turn over and reveal the "key" card to the volunteer.
Slide 7 Analyze the Card Trick Why does this trick work? Discuss
with your group/partner why you think the key card is in the eighth
position on the third layout. What kind of mathematical functions
might be working behind the trick? Slide 8 Finding the Functions
The mathematical explanation relies on finding a function that
accepts as input the current position of the key card and returns
as output the new position of the key card, after the cards have
been picked up and laid out. Each successive round of laying out
the cards narrows the possibilities, until the card must be in a
specific spot. Slide 9 There are two functions at work in this card
trick. To find the first function that locates the key card, assign
each position in the piles a number from 0 to 14 and assign each
row of the layout the numbers 0 to 5. Positions Pile 1 Positions
Pile 2 Positions Pile 3 Row Numbers 0120 3451 6782 910113 1213144
Slide 10 From the chart, find the function that gives the row in
terms of the current position; call this function R(c), where c =
current position and R(c)= row of current position Input: current
position; Output: row of current position On your worksheet fill in
the values of this function and then find the function in terms of
c and r. Chart the values of the function Slide 11
R(0)=___R(1)=____ R(2)=___ R(3)=____R(4)=___ R(5)=___
R(6)=____R(7)=___R(8)=___ R(9)=____R(10)=___R(11)=___
R(12)=____R(13)=___R(14)=___ Current PositionRow 0120 3451 6782
910113 1213144 Given the current position, what row is the card in?
Slide 12 Notice that input is an integer and the output is also an
integer. This is a special type of function in which the Range is a
series of integers whose graph appears in steps. What type of
function is this? Slide 13 R(c) is a Greatest Integer Function:
Slide 14 The next function finds the new position given the row the
original card was in. Chart the new position of the key card by
studying the link to its row. Fill in the values of this function
on your worksheet. Input: row of current position; output new
position of card Slide 15 Given the row, what is new position?
Write the function that gives the new position in terms of the
original row. Choose a key pile to explore the new positions. This
will be the function N(r)=new position Original Layout Row New
Positions 0120 3451 6782 910113 1213144 Slide 16 New Positions of
the cards in the Key Pile on Layout 1 : Given Row 0, 1, 2, 3, or 4,
the Key Card will go to a position ____, _____, ____, _____, or
_____. What is the Domain and Range of this function? New Position
Layout 1 012 34A BCD E1011 121314 Current Position Layout 0 Row
0120 3451 6782 910113 1213144 Slide 17 After the first pick up and
layout, what happens to the key pile? Current Position row number
new position 0 1 2 0 5 3 4 5 1 6 6 7 8 2 7 9 10 11 3 8 12 13 14 4 9
Given an input of the row, find the new position. Write this
function. What type of function is this? Slide 18 Write the Rule
for N(r)=_____? The new position depends on the original row
number. What is the function rule that gives the new position as a
function of the row number: n(r)=_________? RowNew Position 0 1 2 3
4 Slide 19 N(r) is a Linear Function: Go to your Nspire and define
the function N(r) Slide 20 Analyzing the Card Trick First Function
finds the row given the current position. Second function finds the
new position given the row. Slide 21 We need a function that would
allow us to input the current position, find the row, then use the
row number to find the new position? What type of function or
function operation would that be? Slide 22 The Composite Function:
N(R(c)): The actual function that determines the location of the
card is the composite function N(R(c)). Input the function that
finds the row of the current position into the function that gives
the new position from the input of the row. R(c)=int(current
position/3) gives row of current N(r) = row number +5 gives new
position from row N(R(c)) gives new position from row of current.
Slide 23 Explore Iterations of the Composite Function to model the
3 layouts of the cards! Your Composite function tracks the new
position of the key card. Input current position of cards, output
is the new position of the key pile cards. Find 3 iterations of the
composite function. Slide 24 The First Iteration N(R(C))= 5 +
int(current position/3) The first iteration, Layout 1 (first layout
after layout 0 yeilds: n(0)=5 n(5)=6 n(10)=8 n(1)=5 n(6)=7 n(11)=8
n(2)=5 n(7)=7 n(12)=9 n(3)=6 n(8)=7 n (13)=9 n(4)=6 n(9)=8 n(14)=9
On the First Iteration, Layout 1, the key card will be in position
5,6,7,8,or 9. Slide 25 The Second Iteration: The second iteration,
Layout 2 of n(c) yields: n(5)=6 n(6)=7 n(7)=7 n(8)=7 n(9)=8 On the
second iteration, Layout 2, the key card will move to a position
6,7, or 8. Slide 26 The Third Iteration: The third iteration,
Layout 3 of this function yields: n(6)=7 n(7)=7 n(8)=7 *** on the
third iteration, the key card move to position 7, the "eighth" card
in the new stack. Slide 27 Excel, Ti-Nspire, or Ti-84 Looking at a
spread sheet and graph will help visualize the functions. Choose
your graphing technology to see another representation of the card
trick On the Nspire -Label four columns as first, second, third,
and final. Define the first column in the grey cell using the
sequence command: seq(n,n,0,14) Enter the composite function
=5+int(first/3) in the grey column for the second column. Do this
for the third and final columns. Slide 28 Excel On Excel-label the
columns, go to the first blank cell, A2, and type 0,1,2,then drag
to fill in the values to 14. In the second column, excel recognizes
the INT function, press = 5+int(A2/3). Then drag to fill in the
rest of the cells. Do this for column 3 and the final column. To
graph, have cursor over a cell and enter insert graph. Choose the
best model that represents the card trick. Slide 29 TI-84/83 Press
Stat; Edit Go into the heading of L1 Press 2 nd STAT; Go over to
OPS: find sequence command: seq( Then enter x,x,0,14) Press enter:
the values from 0 to 14 should then fill Move to heading of L 2 and
enter y 3 (L 1 ) or you can directly enter the composite function
Slide 30 TI-84/83 Press enter and then move to the heading of L3
and enter the same expression, changing L1 to L2. L3 represents the
values of the composite function on the second layout after t
choosing the key pile. Do the same in L4, this will be the third
layout. Slide 31 Visualizations from the TI-Nspire Slide 32
Extensions: What happens if the key pile is always picked up last?
What if the key pile is always picked up first? What if we vary the
order of the piles-picking up key pile first, then second, then
third?-Does the key card always end up in a unique position? What
if we use 21 cards with three columns?
