+----+----+----+----+
| 23 | | | |
+----+----+----+----+
| | | 64 | |
+----+----+----+----+
| | N | | |
+----+----+----+----+
| | | |101 |
+----+----+----+----+
In the 4x4 grid provided above:
- Each the 16 values appearing in the 16 cells is a positive integer.
- The 4 values corresponding to each of the 4 rows are in arithmetic sequence.
- The 4 values corresponding to each of the 4 columns are in arithmetic sequence.
Determine the total number of
distinct positive integer values that N can assume.
(In reply to
Solution by Larry)
Using Larry's notation, we want to count the number of solutions.
All numbers need to be positive, but it is sufficient for all corners to be positive. In other words, we need H and Q to both be positive.
H = 23 + 3a, so 23 + 3a > 0, or a > -7.666
As for Q, we can calculate it as follows:
G= 23 + 2a and K = 64
Because this is an arithmetic sequence,
S = G + 3*(K-G) = 146 - 4a
Then Q = 101 + 3*(S-101) = 236 - 12a
Q > 0 if a < 236/12 = 19.666
-7.666 < a < 19.666 means that there are 27 valid integer values of a, and that is our answer.
Edited on February 13, 2023, 9:24 am
Counting analytically
