Question 1131667: The positive integers are arranged in the pattern illustrated below:
1 2 5 10 17 26
3 4 7 12 19 28
6 8 9 14 21 30
11 13 15 16 23 32
18 20 22 24 25 34
27 29 31 33 35 36
If this pattern continues indefinitely, what is the number immediately above 39863?
Notice that the 30,28,26 are all in the 32,34,36 column, the 17,19,21 in the 23,25,35 column and so on.
Answer by greenestamps(13195)   (Show Source):
You can put this solution on YOUR website!
A very interesting and unusual problem; thank you for giving us something out of the ordinary to think about.
The pattern is a square array; here is the beginning of the pattern:
1 2 3 4 5 6
+-----+-----+-----+-----+-----+----+
1 | 1 | 2 | 5 | 10 | 17 | 26 |
+-----+ | | | | |
2 | 3 4 | 7 | 12 | 19 | 28 |
+-----------+ | | | |
3 | 6 8 9 | 14 | 21 | 30 |
+-----------------+ | | |
4 | 11 13 15 16 | 23 | 32 |
+-----------------------+ | |
5 | 18 20 22 24 25 | 34 |
+-----------------------------+ |
6 | 27 29 31 33 35 36 |
+----------------------------------+
Here is the pattern in which the positive integers are put into the array:
(1) The squares in the array:
The integer 1 is in the top left corner in a 1x1 square.
The integers 1 through 4 are in the top left corner in a 2x2 square.
The integers 1 through 9 are in the top left corner in a 3x3 square.
...
The integers 1 through n^2 are in the top left corner in an nxn square.
(2) The integers on the main diagonal of the array:
The integer 1^2=1 is in position (1,1).
The integer 2^2=4 is in position (2,2).
...
The integer n^2 is in position (n,n).
(3) The integers above n^2 in column n:
The integers in column 3 above 3^2=9 are the odd numbers greater than 2^2 and less than 3^2.
The integer in column 4 above 4^2=16 are the even numbers greater than 3^2 and less than 4^2.
...
The integers in column n above n^2 are either the odd numbers greater than (n-1)^2 and less than n^2, if n is odd; or the even numbers greater than (n-1)^2 and less than n^2, if n is even.
(4) The integers to the left of n^2 in row n:
The integers to the left of 3^2=9 in row 3 are the even integers greater than 2^2 and less than 3^2.
The integers to the left of 4^2=16 in row 4 are the odd integers greater than 3^2 the and less than 4^2.
...
The integers to the left of n^2 in row n are either the even integers greater than (n-1)^2 and less than n^2, if n is odd; or the odd integers greater than (n-1)^2 and less than n^2, if n is even.
SO....
How do we answer the question: what number is above 39863 in the array?
(1) The square root of 39863 is between 199 and 200. So 39863 is in either row 200 or column 200. But the integers in row 200 to the left of 200^2=40000 are odd and the integers in column 200 above 40000 are even; so 39863 is in row 200 of the array.
(2) We need to find the number in row 199 directly above 39863. To do this, observe the following:
There is a common difference of 7 between the first few entries in rows 4 and 5.
There is a common difference of 9 between the first few entries in rows 5 and 6.
...
There is a common difference of (2n-3) between the first few entries in rows (n-1) and n.
So the common difference between the first several integers in columns 199 and 200 is 397.
And that gives us the answer to the problem: The integer above 39863 in the array is 39863-397 = 39466.
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