Problem Statement: How many eggs were broken? (And is there more than one answer?)
Process: Given that she lined them up by twos and one was left over, by threes and one was left over, by fours and one was left over, by fives and one was left over, by sixes and one was left over and by sevens and it came out evenly, we figure the number had to be a multiple of seven and end in a one or a six. (Anything divisible by five with one left over has to end in a one or six: 6, 11, 16, 21, etc. We figure that out when we eliminated 49 as an answer.)
So we started with 21 (the first multiple of seven that ends with a one or a six) and found you couldn't divide by three and have one left over. We went up by sevens until we found 56 (but when we divided by two we didn't have one left over) and 91 (but when we divided by 4 we didn't have one left over) and 136 (which didn't work for two) until we figure out we could go up by 35 to reach the next multiple of seven that ends with a one or a six.
Going up by 35's (161, 206, 231, 276, …) finally led us to 721. An answer that worked! Everyone in the group double-checked it and we found an answer. Now is it the only answer. We realized that the answer had to be odd (because any number divided by two with one left over is odd) so we could go up by 70's.
So we started a new list (791, 861, 931, 1001…) until we found 1141 worked. Nikki the had the brilliant idea to check this pattern and see if the next answer would be 420 more (1141 + 420 = 1561) and it did. So did 1981 (1561 + 420) and 2401 (1981 + 420).
My question is why doesn't 721- 420 (301) work? I checked it and it worked! How did we miss that?
Solution: 301 and 721, 1141, 1561, 1981, 2401. If I'm the insurance agent I only pay for the 301 and tell her to get a better cart because the solutions will keep getting larger and larger
Extension: I was wondering why 420 was the magic number so I...