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There are 4950 possible pairs in the initial problem statement. Sandy gets one of 197 possible sums, and Peter gets one of 2,869 possible products. Of those 2,869 products, 1,765 can be produced with only possible pair of numbers: something like 67 can only be (1, 67), whereas 240 could be (3, 80) or (5, 48) or (4, 60) or 5 other possible pairs. Peter doesn't know the answer, so when he tells that to Sandy, she learns that it can't be (1, 67) but it still could be (3, 80) or the like. Before Peter told Sandy he didn't know, only 4 sums could have been caused by a unique pair (198, 3, 2, and 197). Peter telling Sandy he doesn't know lets her rule out lots of pairs, and after doing so, there are 9 sums that would have a single pair remaining that hadn't been ruled out. For example, were the sum 165, Sandy could have concluded that the only possible pairing would be 69 and 96, since the other pairs that add up to that number (e.g., 74 and 91, 80 and 85, etc.) would have unique products that Peter would have known about. That she doesn't know the answer yet therefore tells Peter that 69 and 96 is not a possible pair. Were the product 6624, Peter would now know that the only possible remaining pair was 72 and 92, and he would know the answer. But since he didn't know the answer, now Sandy knows that it can't have been 72 and 92 either. This crossing out continues until Peter realizes that 70 and 96 was not a viable pair, which lets him realize that the only other way to get 6720 was to have the numbers be 80 and 84, and he declares he knew the answer. [Assuming I got the correct number of rounds] |
Round #1: ... (many)
Round #2: ... (many)
Round #3: (1,4), (72,92) and (72,98)
Round #4: (2,3), (80,90)
Round #5: (1,6), (75,96)
Round #6: (72,99)
Round #7: (81,88)
Round #8: (70,99)
Round #9: (77,90)
Round #10: (72,95)
Round #11: (76,90)
Round #12: (70,96)
Round #13: (90,84)
Round #14: (66,98)
Round #15: Solution is "77" and "84"