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# Counting

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John counts up from 1 to \(18\), and then immediately counts down again to 1, and then back up to \(18\), and so on, alternately counting up and down: What is the 1000th integer in his list?

Apr 17, 2022

#1
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Is it like this:

\(1,2,\cdots,18,17,16,\cdots,2,1,2,\cdots,17,18\cdots\)

or like this:

\(1,2,\cdots,18,18,17,16,\cdots,2,1,1,2,\cdots,17,18\cdots\)?

Apr 17, 2022
#2
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1,    2,    3,   4,    5,   6,   7,   8,   9,  10,  11,  12,  13,  14, 15, 16, 17, 18

35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19

The 1000th count should land on: 1000 mod (35 - 1) ==14th number counting from 1 at the left.

Here are the last 30 numbers counted by computer !!

(970, 18)
(971, 17)
(972, 16)
(973, 15)
(974, 14)
(975, 13)
(976, 12)
(977, 11)
(978, 10)
(979, 9)
(980, 8)
(981, 7)
(982, 6)
(983, 5)
(984, 4)
(985, 3)
(986, 2)
(987, 1)
(988, 2)
(989, 3)
(990, 4)
(991, 5)
(992, 6)
(993, 7)
(994, 8)
(995, 9)
(996, 10)
(997, 11)
(998, 12)
(999, 13)
(1000, 14)

Apr 17, 2022