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Texas Instruments International Conference Chicago, IL 2012 Adapted from NCTM Mathematics Teacher, March 2011, "AN EXCEL-LENT CARD TRICK" by Holly S. Zullo.

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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.

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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 :

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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

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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.

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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.

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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?

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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.

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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

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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

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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 Given the current position, what row is the card in?

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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?

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R(c) is a Greatest Integer Function:

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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

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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

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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 A BCD E Current Position Layout 0 Row

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After the first pick up and layout, what happens to the key pile? Current Position row number new position Given an input of the row, find the new position. Write this function. What type of function is this?

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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

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N(r) is a Linear Function: Go to your Nspire and define the function N(r)

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Analyzing the Card Trick First Function finds the row given the current position. Second function finds the new position given the row.

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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?

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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Visualizations from the TI-Nspire

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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?

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