The Magic Windmill
I named the following magic figure “The Magic Windmill”, as it looks like a toy plastic windmill with 6 blades.
This magic figure has 18 triangular cells. Each cell contains one of the consecutive numbers from 1 to 18. The example shown here has a, magic constant of 57, which is one third of the sum of all the numbers contained in the windmill (1+2, … +18 = 171).
The rows indicated by arrows add up to 57. There are two rows of 6 cells for each of the three directions. They are:
Horizontal (blue arrows): 6+15+11+8+7+10 = 57; 9+12+14+4+5+13 = 57;
Slanted right (green arrows): 1+6+15+14+4+17 = 57; 2+11+8+5+13+18 = 57;
Slanted left (gray arrows): 12+14+15+11+2+3 = 57; 16+17+4+5+8+7 = 57;
Furthermore, the same magic constant of 57 is manifested in more ways:
- The sum of the three top cells and the three bottom cells: 1+2+3 + 16+17+18 is also 57.
This is true for the other two directions as well:
1+6+9 + 10+13+18 = 57; 3+7+10 + 9+12+16 = 57;
All of the above are linear symmetries. But we can also identify point symmetries, with respect to the center:
- The sum of all numbers in the central hexagon (15+11+8+5+4+14) is 57
- The sum of all numbers adjacent to the central hexagon (green cells: 6+2+7+13+17+12) is 57
- As a consequence, the sum of all numbers in the windmill tips (1+3+10+18+16+9) is 57
Opposite cells
Now, if we imagine the 18 numbers as nine groups of 2, then since 171/9 is 19, we should find couples of triangles with numbers adding up to 19.
Since the whole windmill is symmetrical, most likely these couples will also follow some symmetry.
We can subdivide our figure into three levels of triangles, with respect to the center: The triangles within the central hexagon, the green triangles adjacent to them, and the tips of the windmill.
Now, we notice that each couple of second level green triangles, opposite to each other with respect to the center (2 and 17, 7 and 12, 6 and 13) always add up to 19.
So do the windmill tips: 1 and 18 = 19, 3 and 16 = 19, 10 and 9 = 19.
There are many solutions, such as the example we have been using so far, that satisfy all the previous properties.
However, the solution shown below maintains all of the above properties, but also displays more properties, including the following: The opposite triangles in the center hexagon also contain numbers that in all cases add up to 19.
So far, we discovered nine linear symmetries in three directions, and three point symmetries with respect to the center. Plus we discovered that our solution is balanced with respect to opposite triangles at the same level:
Solutions along linear symmetries:
- horizontal (blue arrow):
8+13+15+5+9+7 = 57; 12+10+14+4+6+11 = 57; 1+2+3+16+17+18 = 57
- slanted (gray arrow):
1+8+13+14+4+17 = 57; 2+15+5+6+11+18 = 57; 3+9+7+12+10+16 = 57
- slanted (green arrow):
10+14+13+15+2+3 = 57; 16+17+4+6+5+9 = 57; 12+8+1+18+11+7 = 57
Solutions according to center point symmetries:
- windmill tips: 1+3+7+18+16+12 = 57
- star points (green): 8+2+9+11+17+10 = 57
- hexagon: 13+15+5+6+4+14 = 57
Opposite triangles:
- windmill tips: 1+18 = 19; 3+16 = 19; 7+12 = 19
- star points (green): 2+17 = 19; 9+10 = 19; 8+11 = 19
- hexagon: 15+4 = 19; 5+14 = 19; 13+6 = 19
All of the above is amazing, but still have a lot to discover! Once we look at a perfectly balanced solution, other properties come to light.
The vertical direction
If we look at the vertical direction, this is obviously rotated 90 degrees with respect to the horizontal, and 30 degrees with respect to the above slanted directions.
We can now consider the cells in this direction and we discover more combinations of cells with numbers adding up to our magic number:
The blue cells aligned with the central vertical axis, added to the left and right tips of the windmill, add up to 57.
The six gray cells aligned vertically, contiguous to the above, three to the right and three to the left, also add up to 57.
The next six green cells aligned vertically, contiguous to the above, three to the right and three to the left, also add up to 57.
If this combination of sums is true vertically, will it be also true for the slanted directions obtained by rotating our axis of symmetry by 60 degrees and by 120 degrees?
Two more slanted directions
To our amazement, the answer to the previous question is yes!
You can verify how the following two orientations give us six more ways to reach the magic sum:
Thus, we can add to our list 9 more solutions along these 3 new directions:
- vertical:
12+2+15+4+17+7 =57; 13+14+16+3+5+6=57; 1+8+10+9+11+18=57
- slanted (left figure):
16+8+13+6+11+3 =57; 14+4+18+1+15+5=57; 12+10+17+2+9+7=57
- slanted (right figure):
1+10+14+5+9+18 =57; 12+13+15+4+6+7=57; 8+2+3+16+17+11=57
As a bonus, the numbers in opposite triangles at 3 different levels, all add up to the same number (19). Of course, 3 x 19 = 57.
Because of this last property, we can add up the numbers of these opposite cells at each level, in each direction, and obtain 57 three more times!
Sums of opposite arms: 1+8+13+6+11+18 = 57; 3+2+15+4+17+16=57; 12+10+14+5+9+7 = 57.
The perfect solution of the Magic Windmill shown above displays a total of 24 ways in which we can add up groups of numbers from the first 18 integers, within the symmetry constraints of our geometrical figure, and achieve the magic sum of 57: It displays 18 linear (or composite) symmetries and 6 center point symmetries.
If you enjoyed reading this article you may also enjoy: The Magic Cross and Other Magic Figures at https://medium.com/@gori70/the-magic-cross-and-other-magic-figures-3c79c88b67ff
