Mathematical Christmas Tree

Stimulated by Johann Weilharter (@jweilharter) and  Clifford  Pickover (@pickover) I wrote some Mathematica code to solve the following Christmas tree puzzle :

XMasTree_1.gif

XMasTree_2.png

Let us define the Xmas tree property :

XMasTree_3.png

XMasTree_4.png

How  many solutions exist?

XMasTree_5.png

XMasTree_6.png

Since solutions can be rotated we only keep solutions with "ordered corner values".

XMasTree_7.png

XMasTree_8.png

How many solutions with ordered corner values exist?

XMasTree_9.png

XMasTree_10.png

Since numbers on the inner part of the edges may be flipped, we remove solutions which are duplicated in this sense.

XMasTree_11.png

XMasTree_12.png

How many solutions do we have now?

XMasTree_13.png

XMasTree_14.png

What are all these solution?

XMasTree_15.png

XMasTree_16.png

What are permissible corner number combinations?

XMasTree_17.png

XMasTree_18.png

XMasTree_19.png

Which (equal) sums for the triangle sides allow solution s?

XMasTree_20.png

XMasTree_21.png

Which corners combinations allow how many essentially different solutions?

XMasTree_22.png

XMasTree_23.png

The Mathematica code is available for downlowad
Created with the Wolfram Language