Kite Rings are of enormous importance for many puzzles based on triangles. As an illustrating example you see a number of DiDom kite rings on this page. For puzzles with figures containing kites (e.g.DiDom), a kite ring in each solution is a must. The ring is determining the only possible positions of the kites and separates areas with different grid orientations. Other puzzles in this category (Stomachion, Drafter, etc.) may have kite rings. In that case you find always a pair of figures for each kite position, one inside and the other outside the ring, forming the kite from two halves. The same pairing occurs, if the number of figures with kites is less than positions on the ring. With the Stomachion there are no kite rings possible, too few figures.
Before we can use the Logelium solver to calculate the DiDom solutions, we must find all kite rings that fit into the outer form. That looks easier than it actually is, because there are many ring types and each one fits usually in many ways into the form. I denote the ring types with letter codes. The code has no special meaning besides just naming.
Generally you can observe that the number of solutions decrease with increasing complexity of the ring structure due to the number of restricting conditions.
These are “simple” kinds of rings. They are composed from squares of area five. The vertex points of the squares are shared by both grids. The ring can be either located completely inside of the form or can share some part of the border of the form. Sometimes this results in cutting the primary grid into several parts.
With a little practice and help of the grid diagram on the left of each example you can see the kite ring structure in the solutions.
Grids and Kite Rings |
Solution Example |
Number Solutions |
Ring Angle |
Ring Code |
4643 |
+ λ |
A |
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101 |
- λ |
B |
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2375 |
- λ |
C |
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18 |
+ λ |
D2 |
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1 |
+ λ |
L4 |
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97 |
+ λ |
H4 |
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5 |
+ λ |
W7 |
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61 |
+ λ |
I2 |
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8 |
- λ |
T2 |
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32 |
- λ |
LL1 |
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none |
+ λ |
NN2 |
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none |
+ λ |
HH2 |
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27 |
+ λ |
LT2 |
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none |
+ λ |
UU2 |
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17 |
+ λ |
VV2 |
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17 |
+ λ |
WW6 |
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none |
+ λ |
TT1 |
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none |
+ λ |
RR2 This ring configuration is obviously impossible to solve, because there is no tiling even with Doms. |
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none |
- λ |
U2 The U-shaped part is the secondary grid, the other two parts belong to the base grid. |
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752 |
- λ |
QQ3 |
If a kite ring is large enough, another one can be placed inside. This leads to recursive ring structures. The recursion is finite, because the inner rings always get smaller.
none |
- λ |
QQ3 und BY This is the only recursive combination of kite rings found, which has a chance for a solution with DiDoms. So its a bit of a pity that there are actually no solutions. All three grids have different tilt angle. The picture on the right shows all possible kite positions. |
The solutions of configurations with multiple kite rings are connected with solutions of the single rings. The reason is the local symmetry of the rings. Mirror symmetrical rings (e.g. type A or B) can be flipped around so that they lay in the base grid. If the ring contains no kites, you find a related solution of the other ring by flipping the partial solutions contained in the ring.
That means also that such multiple kite ring configurations can have no solution, if all single ring relatives have no solution.
Grids and Kite Rings |
Solution Example |
Number Solutions |
Ring Angle |
Ring Code |
16 * 2 |
- λ |
Two times AY Beautiful symmetrical form with two kite rings. |
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145 |
+ λ |
AX and BX The kites obviously have to live in one or the other kite ring. We always have a local symmetry in the ring without whole kites. You can rotate the interior of the ring by 180° getting another solution. If we try to mirror the ring, the ring surprisingly dissolves in the base grid and we get a solution with only one ring left. |
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16 |
+ λ |
AX and BY This configuration has three different grid angles. |
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2 |
+ λ |
AX and D1Y This configuration has three different grid angles and is the only found with two different rings which have solutions. |