Grades 7-8 Video Solutions 2021-video 2021 7-8/26

video 2021 7-8/26

Video Transcription

Question 26. 2,021 colored kangaroos are arranged in a row and are numbered from 1 to 2,021.  Each kangaroo is colored red, gray, or blue. Among any three consecutive kangaroos, there are always kangaroos of all three colors. Bruce guesses the colors of five kangaroos.  These are his guesses. Kangaroo 2 is gray. Kangaroo 20 is blue. Kangaroo 202 is red. Kangaroo 1002 is blue. Kangaroo 2,021 is gray. Only one of his guesses is wrong. What is  the number of the kangaroo whose color he guessed incorrectly? So if we set this up and have our colors be variables, we could say, for example, kangaroo  will be the color X. Kangaroo 2 will be the color Y. Kangaroo 3 will be the color Z. This satisfies that three consecutive kangaroos have different colors. And then this will  repeat constantly. Kangaroo 4 will have a color of X. This will keep going. Kangaroo 5, a color of Y. Kangaroo 6, a color of Z, and so on. From this, we notice that if we  take kangaroo number unknown divided by 3, then we get a color of Z. For example, kangaroo  3, 3 divided by 3, is 1. And there is no remainder. Or kangaroo 9 divided by 3, again, there is  no remainder, so color Z. Next, if we do kangaroo number unknown divided by 3 with a remainder of 2, we can call that color Y. So kangaroo number 2 can't be divided by 3, and it has  a remainder of 2, so it's color Y. Kangaroo 5 also would have a remainder of 2, so it's a color Y. Finally, we have a remainder of 1. So any kangaroo with a remainder of 1 after  being divided by 3 will have a color of X. Kangaroo 1, 4, 7, and so on.  With this information, we can start to look at the guesses. So we have the guesses 2, 20, 202, 1002, and 2021.  We just have to divide these by 3 and see what the remainders are. 2 divided by 3 gives us a remainder of 2, so the color will be Y. 20 divided by 3 also gives us a remainder  of 2, so a color of Y. 202 divided by 3 is a remainder of 1, so it's color X. 1002 divided  by 3 has a remainder of 0, so it is color Z. And finally, 2021 divided by 3 has a remainder of 2, so it is color Y. If we compare these to the guesses of gray, blue, red, blue, and  gray, we notice that red is only for X's and blue is only for Z's. However, we have gray and blue on the Y's. And we notice that since there can only be one wrong guess, then the  second, the blue, will be half to incorrect. So that means that the guess for kangaroo number 20 is incorrect, so our answer is B.

Video Summary

Bruce's guess for kangaroo 20's color is incorrect. The pattern of the kangaroo colors repeats every three with different colors: X, Y, Z. Kangaroo numbers divided by three with a remainder one are color X, with a remainder two are color Y, and with no remainder are color Z. Calculations show kangaroo 2 is Y, 20 is Y, 202 is X, 1002 is Z, and 2021 is Y. Bruce guessed kangaroo 20 as blue (Z), but based on the remainder method, it should be Y. Therefore, the erroneous guess is for kangaroo 20.