Professor Sorites is giving a test to the four members of his logic seminar; all are flawless logicians. "I'm thinking of two digits; each is from 0 to 9 inclusive. I'll give each of you a slip of paper with a clue to the digits; don't let the other three see it. For Annie's clue, I'll multiply the digits and write their product. For Bert, I'll add the digits and write their sum. For Chester, I'll subtract one from the other and write their difference as a non-negative number. And for Dagmar, I'll add Bert and Chester's clues together and divide their sum by Annie's clue." "Annie, now that you've thought about your clue, can you identify the two digits?" She says "Yes, I know what they are." "Good. Write them on your slip and pass them to me. What about you Bert?" "Give me a minute more. Ah, got them!" "Pass them up. Chester?" After thinking hard, Chester scribbles his answer and hands it to the professor. Finally, Dagmar says, "Okay, here's mine." All the students' answers are correct. What are the two digits?
Incorrect. Betty would get the product 14 which can only be gotten from 2 and 7. So she would be certain of a unique answer. Bert however, would get the sum 9, which can be gotten from 5 & 4 or 7 & 2, so Bert cannot be certain.
Incorrect. Annie would have a 9 on her paper, but that means the digits could also be 3 & 3, so she can't be certain.
one of the digits cannot be 0, because we need annie's clue in the denominator for dagmar's clue; and because of bert's clue, the numbers will need to be very minimal it's probably 1, 2
Not quite. HINT: Spoiler Start by finding all the pairs that give a unique product (i.e. 14 can only be gotten from 2 & 7). Those are the only pairs Annie can be certain about, so it has to be one of those. From that list, find the pairs with a unique sum. Those are the only pairs Bert can certain about. And keep going from there...
why do you progress from Annie then to Bert? why do you suggest that the clues given to students are linear? ... anything over 1 and 2 stop yielding unique sums
2 3 5 7 3 4 6 8 1 2 4 6 Spoiler so, basically, one of the digits has to be one, and the other has to be 2 3 5 7. Dagmar won't know, because under all circumstances, he's gonna get the integer 2 (sucks to be him), so he's gonna cover all his bases with [1,2] [1,3] [1,5] [1,7] annie, bert, and chester can find out their answers pretty easily.
From here, you can simplify quite a bit. The only sums Bert could be certain about would be: 1 (0,1) 2 (0,2) 4 (1,3) 16 (7,9) 17 (8,9) 0 can't be involved, so that eliminates the first two. 4 & 16 would both yield a 2 from the subtraction, so that eliminates those two because the kid wouldn't be able to be certain there. That would leave 17 (8 & 9) as the only combo that would work. I think.
Bert cannot be certain about 2, as it can be 1+1. Neither about 4 (2+2 also possible) - 0 is also a non-negative number, and I don't see where it states explicitly that the numbers have to be different, so same numbers, e.g. 9,9 should also be possible. Bert could be certain about 1 (0,1) or 17 (8,9) or 18 (9,9) only. It cannot be 0,1 as Alice's clue would come out as 0 and you cannot divide by zero. So it would have to be 8,9 or 9,9. With 9,9, you would get Alice = 9*9 = 81 Bert = 9+9 = 18 Chester = 9-9 = 0 Dagmar = 18/81 = 0.222222222 With 8,9, you would get Alice = 8*9 = 72 Bert = 8+9 = 17 Chester = 9-8 = 1 Dagmar = (17+1)/72 = 0.25 I think both solutions would be correct, or am I missing something?? I mean, 0.25 sounds better as a solution, but how would 9,9 be wrong? I guess I must be missing something, as only one solution can be correct... I can only assume that the correct solution must be 8,9, and that the wording "non-negative number" is misleading, it should be "positive number", because only then you get only one possible solution.