Magic Star Updates


This page, started in March, 2005, will contain material added to this site or links to sub-pages of such material.


Simon Whitechapel A sub-page. With emails starting in 2001, Simon presents solutions for pattern A of magic stars from 15 to 100.
Jon Wharf A sub-page. With emails starting in 2003, Jon confirms the total solution count for all orders and patterns from 6 to11, and provides the total solution count all order 12 patterns. He also supplies some solutions for all patterns of orders 13 and 14.
Andrew Howroyd A sub-page. First contacted me in February, 2005. He also confirms all total solution counts and investigated permutations  between patterns of orders 10 and 11.
Tree Planting Problems A common recreational mathematics puzzle. here it is limited to star shaped patterns.
A Fractional Star This novelty star consists of proper fractions instead of integers.

More short articles will appear on this page as they become available.

Tree Planting Problems

A type of puzzle common in recreational mathematics is referred to as ‘tree planting’. They are so called because they are usually presented as
“A farmer wished to plant x trees in y rows of z trees each. How should he do this.” 

The magic stars may be considered to be tree planting solutions ,with order 5 being 10 trees in 5 rows of 4; order 6 being 12 trees in 6 rows of 4, etc.
My order 5 page shows the 6 patterns possible for 10 trees in 5 rows of 4. One is the magic pentagram, the other five have the same numbers suitable placed so they are magic also. My Polygons and Graphs page has much more on tree-planting problems.

 Here I will present some more complicated star shaped patterns. Interested readers may wish to try assigning numbers so all lines sum the same.

This diagram is the solution to problem 21
in Dudeney’s
Cantabury Puzzles (page 175).


This stylized order 8 contain 4 more spots for numbers. Here we
would use the numbers 1 to 20 arranged so all 18 rows sum correctly.

It has 16 cells to place numbers so ideally would use the numbers 1 to 16 arranged
so the 15 lines all sum to the same value.
Is this possible?




I have not been able to design a star-like
 pattern where the number of trees is less
 then the number of lines.
However, the first diagram can be modified
 to come close, by introducing a third 5 point
 star to give 20 trees in 20 row of 4.




A Fractional Star

This novelty star was found in a grade five exercise booklet and included the following text. [1]  

Add the 4 fractions in each of the six lines.
Are the six sums the same?
Add the 3 fractions in each of the two big triangles.
Are the two sums the same?
Add the 3 fractions in each of the six small triangles.
Are the six sums the same?

[1] Harold D. Larson,  Magic Squares, Circles, Stars: Grade 5.,
16 pp booklet published in 1956.


This page was originally posted March 2005
It was last updated April 03, 2010
Harvey Heinz
Copyright © 1998-2009 by Harvey D. Heinz