Problem number 20. 9 cards numbered from 1 to 9 were placed on the table. Alexa, Bart, Clara, and Deanna each picked up 2 of the cards. Alexa said, my numbers add up to 6. Bart said, the difference between my numbers is 5. Clara said, the product of my numbers is 18. Deanna said, what if my numbers is twice the other one? All four made a true statement. Which number was left on the table? So first, we have to list out all possibilities for these cards. This can be done just by going through the numbers from 1 to 9 and remembering a person cannot have 2 of the same number. If we do this, we get this list. For example, Alexa can only have 1, 5, or 2, 4 because her cards must sum up to 6. Let's assume that Alexa has 2, 4, just for the sake of assumption and we'll see if it leads to a contradiction. If Alexa has 2 and 4, then we have to remove any cards anyone else might have that have a 2 or a 4, because we know each person can only have one card of a certain number, i.e. if Alexa has 2, nobody else can have 2. If we remove all of these cards, we're left with the following. But notice that here, Clara and Deanna both have the exact same pair of cards left over. 3, 6. But they cannot both have 3, 6, because only one of them can have 3. And so this is a contradiction. So this means that our original assumption that Alexa had 2, 4 is wrong. And that means that Alexa must have 1, 5. Now, if we know Alexa must have 1, 5, we can do the same logic and eliminate any cards anyone else has that have a 1 or a 5. In this case, that eliminates these two pairs. Now, let's try again. Since Clara only has two possibilities, let's say Clara has 2, 9 and see what happens. Well, that means that again, we should eliminate any cards that have a 2 or a 9, and we're left with this. But now if we notice Bart and Deanna, Bart must have 3, 8. But that would rule out both of Deanna's pairs, because one of them has a 3 and the other of them has an 8. So this is also a contradiction. That means that our original assumption that Clara had 2, 9 is wrong. That means that Clara must have 3, 6. And again, using this, we can rule out two more pairs for Bart and Deanna. Now, notice that Deanna must always have the card numbered 4, because those are the only possibilities left. Regardless of if she has 2, 4 or 4, 8, she must have the card numbered 4. This means that Bart cannot have 4, 9, because Deanna must have a 4. And that means that Bart must have 2, 7. And now, going backwards, since we know Bart must have 2, 7, Deanna cannot have 2, 4, because Bart already has the card numbered 2. That means that Deanna must have 4, 8. And now we know all of the cards everyone has. Alexa has 1, 5. Bart has 2, 7. Clara has 3, 6. And Deanna has 4, 8. And the only number from 1 to 9 that is not listed among those is the number 9. So the answer is E, 9.

card picking

logical deduction

number puzzle

combinations

process of elimination