These days, I often do some fun maths with my kids at home. I believe my children are challenged by brain teasers and logic puzzles which promote their problem-solving and critical thinking abilities. And, it allows them to discuss topics which may have been too challenging before. One such puzzle is the Perimeter Magic Triangle. This puzzle uses the same concept as the magic square, but requires you to arrange the numbers in a triangle where all three sides add up to the same number.
Cut the numbered circular discs from the printout. The paper with an image of the triangle is the puzzle board.
Solving the magic triangle of order 3
To solve the puzzle you need to arrange the numbered discs from 1-6 into the six empty circles along the sides of the triangle so that the three numbers on each side of the triangle adds to the same total.
There can be multiple solutions to the magic triangle puzzle. Can you figure all of them? What should be the magic square sum of a magic triangle of any order with a given arithmetic sequence ? Is there any strategy to follow in deciding which numbers will be placed on the vertices of the triangle ?
Give it a try and click below to check your solution.
Each magic triangle has multiple solutions. For example, for a magic triangle with arithmetic sequence 1 to 6 the magic sum will be 9, 10, 11 and 12.
The magic sum ( ie sum of all three numbers in each side ) of a magic triangle for any arithmetic sequence can be calculated using the below formula.
Using the following notation,
n = the number of integers per side ( the order of the triangle )
d = difference between the two numbers in the arithmetic sequence
f = first number of the arithmetic sequence
The formula for smallest magic sum is
Magic Sum = ( ½(n²×d) – (n×d) + f ) x 3 + ½(n x d )
Number of Magic Sums = 3n -5
Therefore, for a triangle for series 1-6 the smallest magic sum will be
Sum = ( ½(3²×1) – (3×1) + 1 ) x 3 + ½(3 x 1 ) = ½( 9 – 6 + 2 ) x 3 + ½(3 ) = 9
and , the number of magic sums will be 3 x 3 – 5 = 4
To solve the magic triangle of order 3 there are two different methods and each method will result into two different magic sum.
1. Clockwise Ascending or Descending
In this approach you fill all the triangle vertices in ascending or descending order of the arithmetic sequence, beginning with the smallest or largest number in clockwise direction, respectively. When all 3 vertices are filled in, you will notice that the remaining circles make a small inverted triangle inside the big triangle. Fill the circles of the small inverted triangle in clockwise direction with the remaining numbers in the arithmetic sequence.
Here is the sequence to follow to populate the triangle
V1-> V2 -> V3 -> Inv-V4 -> Inv-V5 -> Inv-V6
Therefore, the two magic sums are 9 and 12
1 + 6 + 2 = 9 , 3 + 4 + 2 = 9 , 1 + 5 + 3 = 9
6 + 1 + 5 = 12 , 4 +3 + 5 = 12 , 6 + 2 + 1 = 12
2. Corresponding Opposite Circles – Ascending or Descending
All the numbers in this method are filled in such a way that each consecutive number in the arithmetic series is filled in the circle horizontally, vertically or diagonally opposite to the circle of the preceding number. You will get two different set of magic triangle depending on the starting number ( when you start with the smallest number in ascending order or when you start with the largest number in descending order).
This is the sequence to populate the triangle
V1-> V2 -> V3 -> V4 -> V5 -> V6
Hence, there are two more magic sums of 10 and 11
1 + 6 + 3 = 10, 1 + 4 + 5 = 10, 5 + 2 + 3 = 10
6 + 1 + 4 = 11, 6 + 3 + 2 = 11, 2 + 5 + 4 =11
What is my kid learning with this puzzle ?
This simple puzzle of Perimeter Magic Triangle allows you to work on your number skills and reasoning skills by narrowing down the choices and then evaluating each choice by adding the numbers and comparing the results.
Extension – Magic triangle of order 4
The magic triangle of order 4 is similar to the magic triangle of order 3 except that there are nine circular discs numbered from 1-9 and you need to place them strategically into the nine empty circles along the sides of the triangle so that the four numbers on each side of the triangle adds to the same total.