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pset8_soln

# pset8_soln - 6.042/18.062J Mathematics for Computer Science...

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6.042/18.062J Mathematics for Computer Science April 5, 2005 Srini Devadas and Eric Lehman Problem Set 8 Solutions Due: Monday, April 11 at 9 PM Problem 1. An electronic toy displays a 4 × 4 grid of colored squares. At all times, four are red, four are green, four are blue, and four are yellow. For example, here is one possible configuration: ' \$ R B Y R Y B G G B R R G B G Y Y ±² ±² ±² ±² ±² 2 3 4 5 ³´ ³´ ³´ 1 ³´ ³´ & % (a) How many such configurations are possible? Solution. This is equal to the number of sequences containing 4 R’s, 4 G’s, 4 B’s, and 4 Y’s, which is: 16! (4!) 4 (b) Below the display, there are five buttons numbered 1, 2, 3, 4, and 5. The player may press a sequence of buttons; however, the same button can not be pressed twice in a row. How many different sequences of n button-presses are possible? Solution. There are 5 choices for the first button press and 4 for each subsequence press. Therefore, the number of different sequences of n button presses is 5 4 n 1 . · (c) Each button press scrambles the colored squares in a complicated, but nonran- dom way. Prove that there exist two different sequences of 32 button presses that both produce the same configuration, if the puzzle is initially in the state shown above. Solution. We use the Pigeonhole Principle. Let A be the set of all sequences of 32 button presses, let B be the set of all configurations, and let f : A B map each sequence of button presses to the configuration that results. Now: > 16! > | B | A | > 4 32 |

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� � � � � � � � 2 Problem Set 8 Thus, by the Pigeonhole Principle, f is not injective; that is, there exist distinct el- ements a 1 , a 2 A such taht f ( a 1 ) = f ( a 2 ) . In other words, there are two different sequences of button presses that produce the same configuration. Problem 2. Suppose you have five 6-sided dice, which are colored red, blue, green, white, and black. A roll is a sequence specifying a value for each die. For example, one roll is: ( 3 , 1 , 4 , 1 , 5 ���� ���� ���� ���� ���� ) red green blue white black For the problems below, you do not need to simpify your answers, but brieﬂy explain your reasoning. (a) For how many rolls is the value on every die different? Example: (1 , 2 , 3 , 4 , 5) is a roll of this type, but (1 , 1 , 2 , 3 , 4) is not. Solution. The number of such rolls is 6 5 4 3 2 · · · · (b) For how many rolls do two dice have the same value and the remaining three dice all have different values? Example: (6 , 1 , 6 , 2 , 3) is a roll of this type, but (1 , 1 , 2 , 2 , 3) and (4 , 4 , 4 , 5 , 6) are not. Solution. There are 5 2 possible pairs of rolls that might have the same value and 6 possibilities for what this value is. There 5 4 3 possible distinct values for the · · remaining three rolls. So the number of rolls of this type is 5 6 5 4 3 2 · · · · (c) For how many rolls do two dice have one value, two different dice have a second value, and the remaining die a third value?
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