Positive Differences
Size:
Mode:
Enter numbers in the bottom row — differences auto-compute upward
Exploring the Triangle
1
Start in Explore mode. Enter
5, 8, 1 in the bottom row. What appears at the top? Try different sets of three numbers.2
Can you find three starting numbers where the top number is 0? How many ways can you do this?
3
What is the largest possible top number if the bottom row only uses single-digit numbers?
The 1 to 6 Challenge (3 rows)
4
Switch to 1 to N Challenge with 3 rows. Can you place the numbers 1 to 6 so each cell is the positive difference of the two below?
5
Can 6 go at the top? Why or why not?
Hint: think about what two numbers from 1–5 could have a difference of 6.
6
Prove that 6 must be in the bottom row. Where can 5 go? Where can 4 go?
7
How many different solutions are there? Use the Solve button to check. Can you find them all by hand first?
The 1 to 10 Challenge (4 rows)
8
Switch to 4 rows. This is much harder. Work in a group and discuss where the largest numbers must go.
9
Can 10 go at the top? Can it go in the second row? Where must it be?
10
Which numbers cannot be at the top? Make a list and explain why for each one.
11
The clue triangles show partial solutions. Can you complete each one? Which is easiest to solve from the clues given?
The 1 to 15 Challenge (5 rows)
12
Switch to 5 rows. There is only one solution (ignoring reflections). Can you find it?
13
Start by figuring out where 15, 14, and 13 must go. What constraints do they create?
Proof and Reasoning
14
In a 3-row triangle, the top number equals
|a − |b − c|| where a, b, c are the bottom row. Can you prove this?15
For the 1–6 challenge, prove that 3 cannot be in the bottom row.
Hint: if 3 is in the bottom row, think about what must be above it and what numbers are left.
16
Is there always a solution for the 1-to-N challenge for any triangle size? What about 6 rows (1 to 21)?
Extensions
17
What if you used sums instead of differences? Enter numbers in the bottom row in Explore mode and see how the triangle grows. How is this related to Pascal’s Triangle?
18
What if the bottom row had 4 numbers (making a wider-based triangle)? Design your own positive differences puzzle.
Positive Differences
The Rules
Each cell contains the positive difference of the two cells directly below it. For example, if the two cells below contain 5 and 8, the cell above contains |5 − 8| = 3.
Explore Mode
Type numbers into the bottom row and watch the triangle auto-compute upward. Try different starting numbers and see what happens at the top.
1 to N Challenge
Place the numbers 1 to N (where N is the total number of cells) so that every cell is the positive difference of the two below it. Each number must be used exactly once. This is a real puzzle — can you solve it?
Tips
Check validates your current arrangement.
Solve finds all solutions (teacher tool).
Print generates blank triangles for classroom use.
Solve finds all solutions (teacher tool).
Print generates blank triangles for classroom use.