What number completes the sequence?

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The discussion centers around a 4x4 matrix with a missing number, where participants explore patterns and relationships among the numbers. The key finding is that the missing number in the matrix is 0, derived from the observation that each number is the product of the digits of the number directly above it in the same column. For example, the missing number at position (4,3) corresponds to the number 210 at (3,3), calculated as 2*1*0. One participant, Zetep, shares that the method was inspired by recognizing that the numbers in the first column followed a pattern related to cubes, leading to the discovery of the rule applied to the entire matrix. The conversation highlights the importance of identifying numerical relationships and patterns in solving matrix puzzles.
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Shown below is a 4 by 4 matrix:

999 998 997 996
729 648 567 486
126 192 210 192
12 18 ? 18

So far, I've not come up with the 'missing' number, but in the process of attempting to develop some sort of 'magic' solution, I've found some interesting relationships with the other numbers.

For example, the average of a(1,2) and a(1,4), 998 and 996, is the a(1,3) number 997. Then, the average of a(2,2) and a(2,4), 648 and 486, is the C(2,3) number 567.

There seems to be other relationships, i.e., a(3,2) plus a(4,2) equals a(3,3). So does
a(3,4) plus a(4,4) equal a(3,3). Plus other similar associations.

Evidently, my once keen mind is not so keen these days with being able to solve this 'mysterious' matrix. Any counsel, advice and/or assistance would be seriously appreciated.

Zetep
 
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The missing number is 0. Each number in the matrix is the product of the digits of the number in the previous row but the same column of the matrix. So the missing number (4,3) is product of the digits of (3,3) or 210 = 2*1*0 = 0.
 
Found The Missing Number

james1234567890 said:
The missing number is 0. Each number in the matrix is the product of the digits of the number in the previous row but the same column of the matrix. So the missing number (4,3) is product of the digits of (3,3) or 210 = 2*1*0 = 0.

Thanks for your reply ... very much appreciated. Obviously, the solution is straightforward when the method you described is taken. I'm curious to know how it was that you chose to apply your approach. Were there 'clues' given the matrix numbers that appeared visible to your keen eye?

Zetep
 
Thanks a lot for your appreciation. This idea struck me when I looked at the first two numbers in the first column i.e. 999 and 729. I remembered that 729 is 9 cubed i.e. 9 * 9 * 9. To my surprise, the number below also follwed the same convention i.e. 126 = 7 * 2 * 9. Thereafter I tried to apply this rule to all numbers in the martrix and it turned out to be correct.
 
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