*Magic Circle and Triangle*.
重新排列数字 1-9；
使得圆圈中8个数字的总和可以被三角形中3个数字的乘积整除。
Rearrange numbers 1-9;
such that the sum of the 8 numbers in the circle is divisible by the product of the 3 numbers in the triangle.
（中学数学必修） （图中显示其中一组答案之一🙏😃）
4 5
6 7
8 9
1 2
3
1+2+3+.....+9=(9x10)/2=45
No . Sum of products
taken . 8 No's 3 No's
Out. in circle in triangle
3. 42. 3x1x2 3x1x7 3x2x7
5. 40. 5x1x2 5x1x4 5x1x8 5x2x4
9. 36. 9x1x2 9x1x4
Magic Triangles
数字1-9
排列组合成一个三角形
每边四个数字的和皆相等
请问有多少个答案？😃🙏🙏
1+2+3+.....+9=(9x10)/2=45
边和至少：
（45+1+2+3）/3 = 17
边和至多：
（45+7+8+9）/3 = 23
边和17：
1
8 9
6 4
2 5 7 3
1
9 7
5 6
2 8 4 3
...........
边和23：
7
3 1
5 6
8 4 2 9
7
6 4
2 3
8 1 5 9
.....
The *4x4 magic square puzzles* is solved by finding the values that make the sums all rows, columns and diagonals equal to the same value. ... The normal variations of these puzzles (in other words, 4x4 puzzles that contain only 1-16 in their cells) have a magic constant of 34 no matter how the numbers are arranged.
*Solutions: 平衡法*
Sum= 1+2+3+..+16
=(1+16)x16/2
=136
Average=136/4=34
1. 2. 3. 4.
5. 6. 7. 8
9. 10. 11. 12
13. 14. 15 16
对换
1. 14. 15 4.
8. 6. 7. 5
12. 10. 11. 9
13. 2. 3 16
再对换
1. 15. 14 4.
12. 6. 7. 9
8. 10. 11. 5
13. 3. 2 16
大事成矣！😃
*3x3 magic square puzzles* is solved by finding the values that make the sums all rows, columns and diagonals equal to the same value. ...
The normal variations of these puzzles (in other words, 3x3 puzzles that contain only 1-9 in their cells) have a magic constant of 15 no matter how the numbers are arranged.
1.
Sum =1+2+3+....+9 =(1+9)x9/2
=45
45/3=15
2. 15=
9+1+5
9+2+4
8+1+6
8+2+5
8+3+4
7+2+6
7+3+5
6+4+5
Total eight equations
5 appears 4 times: placed at centre
8,6,4,2 appear 3 times: placed at corners
9,7,3,1 appear 2 times: placed at the sides
8. 1. 6.
3. 5. 7.
4. 9. 2.
*One and only one solution.*
*今日数学*
*number x*
Divided by 10 with remainder 9
Divided by 9 with remainder 8
Divided by 8 with remainder 7
Divided by 7 with remainder 6
Divided by 6 with remainder 5
Divided by 5 with remainder 4
Divided by 4 with remainder 3
Divided by 3 with remainder 2
Divided by 2 with remainder 1
*What are these numbers?*
*数字x*
除以 10 余数为 9
除以 9 余数为 8
除以 8 余数为 7
除以 7 余数为 6
除以 6 余数为 5
除以 5 余数为 4
除以 4 余数为 3
除以 3 余数为 2
除以 2 余数为 1
*这些数字是多少呢？*
*解答*
X=2519
(X+1) must be divisible by 2,3,4,5,6,7,8,9and 10.
Using LCM, (X+1)=2x2x2x3x3x5x7
👍🙂👏👏🤝
x=LCMxn -1
=2520n-1
n=1,2,3.......
