ECE 290 Homework & Lab - Mallard ECE 290: Computer...

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3/29/09 8:32 PM Mallard ECE 290: Computer Engineering I - Spring 2009 - HWK #1 Solutions Page 1 of 7 HWK #1 Solutions SOLUTIONS Problem 1.1 Before we begin, notice that each of the 5 cards has 16 numbers, exactly one of which is a power of 2. The numbers are listed in numerically increasing order with the power of 2 displayed as the upper left entry. For convenience, we refer to the card with 1 in the upper left corner as the 1-card, the card with 2 in the upper left corner as the 2-card, etc. a. Add the numbers in the upper left corner of each card the volunteer points to. For instance, if the selected number is 19, the volunteer will point to the three cards that contain 19: the 1-card, the 2-card, and the 16-card. As she points to each card in turn, the mathemagician computes the sum 1 + 2 + 16 = 19. b. Consider the cards arranged in left-to-right order: 16-card, 8-card, 4-card, 2-card, 1-card. 16 8 4 2 1 Every integer between 1 and 31 has a unique binary (base 2) representation. In other words, each can be represented uniquely as a sum of powers of 2. In particular, each number between 1 and 31 has a unique 5-bit binary representation. When the volunteer points out which cards contain her number, she is in fact giving the powers of 2 which constitute her number. For instance, with selected number 19, the volunteer notes that 19 does appear on the 16-card, does not appear on the 8-card, does not appear on the 4-card, does appear on the 2-card, and does appear on the 1-card. This corresponds to the binary number 10011, where 1 indicates the number does appear on the corresponding card and 0 indicates it does not appear. c. To extend the trick to include numbers 1 to 63, you will need to make 6 cards, each with 32 numbers on it. The numbers in the upper left corner are: 1, 2, 4, 8, 16, 32. The numbers on each card are the same as the numbers on the 5-card set, but in addition each card contains the same numbers added to 32. Another way to see this: if the binary representation of a number i has a 1 in the 2 k position, then i appears on the 2 k -card. Number in upper left corner = 1 1 17 33 49 3 19 35 51 5 21 37 53 7 23 39 55 8 24 40 56 9 25 41 57 10 26 42 58 11 27 43 59 Number in upper left corner = 8
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3/29/09 8:32 PM Mallard ECE 290: Computer Engineering I - Spring 2009 - HWK #1 Solutions Page 2 of 7 Card contains all numbers x such that remainder(x/2) = 1 7 23 39 55 9 25 41 57 11 27 43 59 13 29 45 61 15 31 47 63 11 27 43 59 12 28 44 60 13 29 45 61 14 30 46 62 15 31 47 63 Card contains all numbers x such that remainder(x/16) = 8,9,10,11,12,13,14, or 15 Number in upper left corner = 2 Card contains all numbers x such that remainder(x/4) = 2 or 3 2 18 34 50 3 19 35 51
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This note was uploaded on 11/16/2009 for the course ECE 290 taught by Professor Brown during the Spring '08 term at University of Illinois at Urbana–Champaign.

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ECE 290 Homework & Lab - Mallard ECE 290: Computer...

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