Magic Flowers – 2
The sums in a Magic Flower are the same for all straight lines. These Magic Flowers use numbers from 1 to 5.
THE CHALLENGE
Use the numbers from 1 to 9 to make Magic Flowers.
EXPLORATION
Looking at the solutions for Magic Flowers 1 and 2, what do you expect the solutions to be for Magic Flowers with even more lines?
Notes
THE CHALLENGE
This, like many of these Puzzles of the Week, can be attacked by playing around with the numbers until a solution is found. Don’t be tempted to think that a more structured approach is better for your students – a great deal will be learned about the mathematics involved and about problem solving by tenaciously working through many examples. Finding a solution, by any method, is always a wonderful reward.
Most of these “equal sum” puzzles can be attacked by adding up some of the straight lines. In the case of this puzzle, add up the four directions – this will include all the numbers once, plus the central circle’s number three extra times. The sum of 1 to 9 is 45. So, the possible sums of the three lines are 45 + 3×1 through 45 + 3×9. Of those, only 48 = 45 + 3×1, 60 = 45 + 3×5, and 72 = 45 + 3×9 are divisible by 4. Dividing them by 4 tells us that the common sums are either 12 = 48 / 4, 15 = 60 / 4, or 18 = 72 / 4.
Let’s look at those three cases.
Common Sum of 12: The central circle will be 1. Making a sum of 12 with a 1 in the center means the other two numbers add up to 11. So, the four directions are: (2 1 9) – (3 1 8) – (4 1 7) – (5 1 6).
Common Sum of 15: The central circle will be 5. Making a sum of 15 with a 5 in the center means the other two numbers add up to 10. So, the four directions are: (1 5 9) – (2 5 8) – (3 5 7) – (4 5 6).
Common Sum of 18: The central circle will be 9. Making a sum of 18 with a 9 in the center means the other two numbers add up to 9. So, the four directions are (1 9 8) – (2 9 7) – (3 9 6) – (4 9 5).
EXPLORATION
For numbers that go from 1 to 2n-1, the central circle has either 1, n, or 2n-1. The sums will be 2n+2 = 1 + 2 + 2n-1, 3n = 1 + n + 2n-1, and 4n-2 = 1 + 2n-2 + 2n-1.