24 states. Three boys played a game of word in which they each wrote down 10 words. Each boy scored three points if neither of the boys had the same word. Each boy scored one point if only one of the other boys had the same word. No points were awarded for words which all three boys had. When they added up their words they found that they each had different scores. Sam had 19 points which was the smallest score and James had the highest score. How many points did James score? So here we have three boys, Sam which had 19 points, another boy which I will just call Noah who had some number of points that's greater than Sam's, and lastly James which had the most points, greater than Noah and Sam. And this is the value the problem asks us about. So let's review the rules of the game. All three players start out with zero points and each one of them writes down ten words. If the word is unique then the player gets three points. If one other boy wrote down the same word then the player gets one point. Last if all three players wrote down the same word then nobody gets a point. Now in order to simplify the game down since Sam the lowest scoring player is scoring close to the maximum points which is 30, let's instead play a complementary game where at the start you get the maximum number of points and you subtract points if the word is not unique. Overall the scores will be exactly the same. So first let's say if the three players start out with the maximum number of points. So 3 times 10 is 30. They'll still each get to choose ten words and if their word is unique they don't get penalized at all. They get minus zero points. However if two of the boys wrote down the same word then they will each get minus two points. And if three boys wrote down the same word then they will get minus three points. Now with all the rules let's represent the points algebraically. Let's let A represent the number of words every boy had and B the number of words that one other boy has had. So A is going to be the same for every single boy but B is going to be unique for each boy. So this means that the number of points P that each boy had is equal to 30 minus 3A minus 2B. Now let's try and figure out what A and B might be for Sam. So we know that Sam scored 19 points. This would mean that A could be equal to 3 and B could be equal to 1 or A could be equal to 1 and B equal to 4. Not being able to rule out any of these let's just start out by assuming that A equals 3 and B equals 1. Since A is the number of words every boy had that means that everyone has the same value for A so everyone's score would be equal to 21 minus 2 times however many words the other boy had which could be unique to each boy. So in this case B could be equal to 0 or could be greater than 0. So if B is equal to 0 then Noah and James would both have 21 points. This is a problem. The problem told us that they found that they each had a different score so B cannot be equal to 0 since that means that they would have the same score of 21. For B is greater than 0 that would mean that Noah and James have no more than 19 points which is a problem as the problem told us that Sam had 19 points which was the smallest score. With all the options exhausted that means that our initial assumption must be wrong. This means that A has to equal 1 and B has to equal 4. Again A is going to be the same for every person so we can just substitute that in and we get the number of points that each player has. So P equals 27 minus 2 times the number of words that each other boy had which can be unique. So B can be 0, 1, 2, or 3 because if it's 4 then the number of points would be less than or equal to 19 which we cannot have. So for Sam we know that B is equal to 4 and now how do we know which value of B works for Noah and James? Well let's review what B is. It is the number of words one other boy has which means that some other boy must have also had the four words that Sam had. Let's call the B value of Noah and James Delta and Epsilon respectively. The key here is that Delta plus Epsilon must be equal to 4. There are only two values of B which can work here and they are 1 and 3. We cannot use 2 twice because then the number of points would be the same for Noah and James. So Noah would have B equals 3 and James would have B equals 1. So the final number of points for James is 27 minus 2 times 1 which is 27 minus 2. So his final score was 25 points. So the question asked us how many points James score? The answer is 25. Letter E.

game

scoring

uniqueness

algebra

points