3,486 different lines are needed to connect all the phones to each other.
Given that everyone in the town of Skunk's Crossing (population 84) has a telephone, to determine how many different lines are needed to connect all the phones to each other, the following arithmetic progression must be performed:
The first inhabitant will have 83 lines (to each of the other residents of the town), and that number will go down by 1, while there will always be a line that has already been contemplated.
- 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
- 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 155
- 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 = 255
- 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 = 355
- 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 = 455
- 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 = 555
- 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 = 655
- 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 = 755
- 81 + 82 + 83 = 246
- 55 + 155 + 255 + 355 + 455 + 555 + 655 + 755 + 246 = X
- 3.486 = X
Therefore, 3,486 different lines are needed to connect all the phones to each other.
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