8
$\begingroup$

$$ \small \begin{array}{c} 1\\ 1 & 2\\ 1 & 3 & 2\\ 1 & 3 & 2 & 4\\ 1 & 5 & 2 & 4 & 3\\ 1 & 4 & 2 & 6 & 3 & 5\\ 1 & 6 & 2 & 5 & 3 & 7 & 4\\ 1 & 5 & 2 & 7 & 3 & 6 & 4 & 8\\ 1 & 9 & 2 & 6 & 3 & 8 & 4 & 7 & 5\\ 1 & 6 & 2 & 10 & 3 & 7 & 4 & 9 & 5 & 8\\ 1 & 9 & 2 & 7 & 3 & 11 & 4 & 8 & 5 & 10 & 6\\ 1 & 7 & 2 & 10 & 3 & 8 & 4 & 12 & 5 & 9 & 6 & 11\\ 1 & 12 & 2 & 8 & 3 & 11 & 4 & 9 & 5 & 13 & 6 & 10 & 7\\ 1 & 8 & 2 & 13 & 3 & 9 & 4 & 12 & 5 & 10 & 6 & 14 & 7 & 11\\ 1 & 12 & 2 & 9 & 3 & 14 & 4 & 10 & 5 & 13 & 6 & 11 & 7 & 15 & 8\\ 1 & 9 & 2 & 13 & 3 & 10 & 4 & 15 & 5 & 11 & 6 & 14 & 7 & 12 & 8 & 16\\ 1 & 17 & 2 & 10 & 3 & 14 & 4 & 11 & 5 & 16 & 6 & 12 & 7 & 15 & 8 & 13 & 9\\ 1 & 10 & 2 & 18 & 3 & 11 & 4 & 15 & 5 & 12 & 6 & 17 & 7 & 13 & 8 & 16 & 9 & 14\\ 1 & 15 & 2 & 11 & 3 & 19 & 4 & 12 & 5 & 16 & 6 & 13 & 7 & 18 & 8 & 14 & 9 & 17 & 10\\ 1 & 11 & 2 & 16 & 3 & 12 & 4 & 20 & 5 & 13 & 6 & 17 & 7 & 14 & 8 & 19 & 9 & 15 & 10 & 18 \end{array} $$

What is the next line in this number triangle sequence?

What is the $100$th line?

$\endgroup$
7
$\begingroup$

The $n$th line represents the order of elimination in the Josephus problem with $n$ people and $k=2$. In other words, the numbers are written in order, skipping one spot every time and wrapping around.

The next line is 1,19,2,12,3,17,4,13,5,21,6,14,7,18,8,15,9,20,10,16,11.

The $100$th line is 1,51,2,76,3,52,4,99,5,53,6,77,7,54,8,89,9,55,10,78,11,56,12,95,13,57,14,79,15,58,16,90,17,59,18,80,19,60,20,98,21,61,22,81,23,62,24,91,25,63,26,82,27,64,28,96,29,65,30,83,31,66,32,92,33,67,34,84,35,68,36,100,37,69,38,85,39,70,40,93,41,71,42,86,43,72,44,97,45,73,46,87,47,74,48,94,49,75,50,88.

$\endgroup$
0
5
$\begingroup$

For nth line, you place n empty slots in a row. Start filling numbers sequentially starting from 1 into the slots by skipping an empty slot every time you place a number. Whenever you reach the end of the line, you wrap around and keep filling the empty slots keeping the above rule in mind. This will the give next line as:

1, 19, 2, 12, 3, 17, 4, 13, 5, 21, 6, 14, 7, 18, 8, 15, 9, 20, 10, 16, 11

Similarly, the 100th line would be:

1, 51, 2, 76, 3, 52, 4, 99, 5, 53, 6, 77, 7, 54, 8, 89, 9, 55, 10, 78, 11, 56, 12, 95, 13, 57, 14, 79, 15, 58, 16, 90, 17, 59, 18, 80, 19, 60, 20, 98, 21, 61, 22, 81, 23, 62, 24, 91, 25, 63, 26, 82, 27, 64, 28, 96, 29, 65, 30, 83, 31, 66, 32, 92, 33, 67, 34, 84, 35, 68, 36, 100, 37, 69, 38, 85, 39, 70, 40, 93, 41, 71, 42, 86, 43, 72, 44, 97, 45, 73, 46, 87, 47, 74, 48, 94, 49, 75, 50, 88

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.