Answer this math question if you can!
Answers
Answer:
"Two in a line" is an arrangement of two circles in a n × n grid such that the circles lie together making a line in the grid.
1. Refer the first attachment.
2. In 4×4 grid, there will be a lot of arrangements of two circles in the grid. In 2×2 grid, there were 6 arrangements and in 3×3 grid, there were 20 arrangements. The difference is of 14 arrangements. Now, the difference is of 14, hence there will be more difference between the next number and the second number of the series.
3. a) The arrangements in the attachment.
b) The predictions matched, as the arrangements are more than 3×3 grid. And the difference between the arrangement of 4×4 grid and 3×3 grid is more than 14.
4. Now, able to observe the series formed that is 6, 20, 42.....
The difference between the second term of the series and the first term of the series is 14 where is the difference between the third term of the series and the second term of the series is 22. Now, if we observe closely the number of differences increased in both the differences (14 and 22) is 8.
Hence, We can conclude by saying that the series will continue in a way in which the next number of the series will be more than the number before. The number of differences that will increase between the next number and the number before will be 8.
5. For n × n, the equation formed is : 2x×2x+2x.
That is, 4x²+2x.
We can also say in this way, 4x²+2 (for 2×2 grid), 4n²+4 (for 3×3 grid), 4n²+6 (for 4×4 grid)...and so on.
Here, n is related such that in n×n grid, x = n - 1.
6. Let's find the number of arrangements using the equation for 5×5 grid.
➸ 4x²+2x
Now, x = 5 - 1 = 4
➸ 4(4)²+2(4)
➸ 16*4+8
➸ 64+8
➸72 arrangements.
7. Let's check if it's correct.
The series : 6, 20, 42, 72 .....
Now, 72 - 42 = 30.
Also, 42 - 20 = 22.
At last, 30 - 22 = 8.
Hence, the number of differences that increased is the same that is 8.
Hence, this equation or rule works well.
