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When the positive integers are arranged in order, filling in the successive diagonals of an infinite grid from top to bottom, as shown, the integer 41 is in the (5,5) spot. What integer would we see in the (10,20) spot if the rest of the grid were visible?

 

 Jun 1, 2021
 #1
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Using the  sum of  differences

 

1    2    4     7     11

    1    2   3     4

       1    1    1

 

We  have  two  rows of  non-zero  results

The polynomial  generating this  series is   ax^2 + bx  +  c

So   we have  this system

a + b + c   =   1

4a + 2b + c   = 2

9a + 3b + c  =  4

Solving this system  gives    a  = 1/2   b = -1/2   c  =  1

 

So....the  10th term in this row  =    10^2 / 2 - 10/2  + 1   =    50  - 5 + 1  =  46

 

Also   in the 10th  column  the terms  are

 

46    57   69    82    96    111

     11    12   13    14    15

          1       1    1      1

And we  have this  system

a + b + c   =   46

4a + 2b + c   = 57

9a + 3b + c  =  69

Solving this  we get   a =  1/2    b = 19/2    c  = 36

Then the 20th term  is       20^2/2  +  (19)(20)/2  +  36   =   426

 

So   position    (10 , 20)  =      426 

 

 

cool cool cool

 Jun 1, 2021
edited by CPhill  Jun 1, 2021

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