Grades 7-8 Video Solutions 2021-video 2021 7-8/28

Video Transcription

Question 28 In a town there are 21 knights who always tell the truth, and 2,000 knaves who always lie. A wizard divided 2,020 of these 2,021 people into 1,010 pairs.  Every person in a pair described the other person as either a knight or a knave. As a result, 2,000 people were called knights, and 20 people were called knaves.  How many pairs of 2 knaves were there? First we can break off these into pairs. For example, if it is a pair of knights, so one knight with another knight, and knights  always tell the truth, then the first knight will call the other one a knight and the second knight will also call the other one a knight, since they are both telling the truth.  If instead we have a pair that is a knight and a knave, then the knight will say the truth and call him a knave, and the knave will lie and call the knight a knave.  And finally, when we have a knave and a knave, both will lie and call each other knights. Now it is important to read that we saw that 20 people were called knaves, and from looking  at the three types of pairs we have, there is only one time where people are called knaves, when there is a knight and a knave.  Since 20 people were called knaves, we know that there will have to be 10 pairs of knights and knaves. So 10 knights and 10 knaves go to this.  Next we have to recognize that there are 21 knights in the town. We know that one person is excluded from this pairing, so as to have an even number.  Now we've already used 10 of the knights, so there are 11 knights left. Now the only other place knights can go is in a knight-knight pairing, so we have to  lose one of the knights so that there can be an even number. There will be 5 pairs of knights and knights. We also know there will be 10 pairs of knights and knaves.  Finally, we know that there are 1010 total pairs. To find out how many pairs of two knaves are, we just have to subtract 10 pairs and 5 pairs  that we know of knight and knave and knight and knight, and then we get our answer. So 1010 minus 10 minus 5 is our answer, D, 995.

Video Summary

In a town with 21 truth-telling knights and 2,000 lying knaves, a wizard forms 1,010 pairs from 2,020 people, where 2,000 are called knights and 20 are called knaves. To determine pair types, only knight-knave pairs result in someone being called a knave, requiring 10 such pairs. With 21 knights total, 11 remain after accounting for these pairs, leaving 5 knight-knight pairs. The remaining are knave-knave pairs. Therefore, subtracting 10 knight-knave and 5 knight-knight pairs from the 1,010 total pairs leaves 995 knave-knave pairs.

Keywords

knights

knaves

pairs

wizard

truth-telling