Number Puzzle: Same total for every row, column, diagonal cont'd (7x7 and 9x9)

7x7:




9x9:

Number Puzzle: Same total for every row, column, diagonal

Do you remember watching The Legend of the Condor Heroes (射鵰英雄傳) (TVB, 1994) where Guo Jing (郭靖) and severely injured Huang Rong (黃蓉) were at Ying Gu's (瑛姑) hut where Ying Gu challenged them to solve a 3x3 square puzzle by filling in numbers 1-9 to total the same in every row, column, and diagonal? Have you wondered how Huang Rong solve it? I'll try to explain it here.

A 3x3 square:

1) Start with "1" in the center square in the first row. From here, add in numbers in sequence, in a diagonally right up direction.
2) There is no more square to the top right diagonal of square 1, therefore, you put the next number, "2" to the right column last row.
3) To put "3", again, there is no more square to the top right diagonal of 2, therefore, you put it to the row above "2" at the most left column.


4) As you go to the top right diagonal of "3", it is being occupied by "1", therefore you proceed one square down and put "4".
5) There is an empty square to the top right diagonal of "4", therefore, fill "5" into the square, followed by "6" to the top right diagonal of "5".

6) To enter "7", there is no square to the top right diagonal of "6" and also no column to the right of "6". Therefore, you fill "7" in the square below "6".
7) To enter "8", there is no square to the top right diagonal of "7", but there is a row above "7", therefore, you enter "8" in the row above "7", and to the most left column.
8) To enter the last number "9", again there is no square to the top right diagonal of "8", therefore "9" is entered in the column to the right of "8" but to the last row.

Do you get it now? In a 3x3 square, every column, row, and diagonal adds up to 15.

This formula can be used for all odd-numbered squares, e.g., 5x5, 7x7, 9x9, and so on.

Try solving the 5x5 square.

Below is the answer for the 5x5 square.

Multiply a 2-digit Number by 99 cont'd

Continuing from the previous post......

What other patterns can you derive from (a), (f), and (g)?

a) 99 x 37
f) 99 x 55
g) 99 x 19

Did you see any pattern?

Well, the pattern that I see is, when 99 multiply a number that both digits add up to 10, (e.g., 99 x 19, 99 x 28, 99 x 73, 99 x 64), it will produce a symmetrical number, eg. 1881, 2772, 7227, 6336. Having this in mind, I hope it can speed up your mental calcuation.

Have fun

Multiply a 2-digit Number by 99

I have left a few more weeks to complete all my term-papers and to face my final exams. Having stress and anxiety within me, one of my ways to destress myself is to play with numbers. So I have decided to share how to get the answer of a 2-digit number (mn) mulitplying by 99 without a calculator. (Does not apply to 99 x 90 (to 99) )

99 x (mn) = abcd (where abcd stands for each single digit)

9 x (m+1) = ac
9 x (n) = bd

For example, 99 x 25:

we take 9 x (2+1) = 9 x 3 = 27.
Therefore, the places where "a" and "c" stand will be replaced by "2" and "7", which becomes 2b7d.

Next, we take 9 x 5 = 45, which takes "b" and "d", so it will be a4c5

Combining both, we will have 99 x 25 = 2475.

More examples,

99 x 17:
9 x (1+1) = 9 x 2 = 18 (1b8d)
9 x ( 7) = 63 (a6c3)
therefore, 99 x 17 = 1683

How fast can you solve the following?
a) 99 x 37
b) 99 x 68
c) 99 x 78
d) 99 x 24
e) 99 x 59
f) 99 x 55
g) 99 x 19

What other patterns can you derive from (a), (f), and (g)?
Look out for the answer in the next posting...


Acknowledgement: I thank my Pri 5 teacher, Mrs Neo, who taught me this, which I remember up to this day.