Diffy
Type
Steps
—
Best
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Enter four numbers and press Start
A
B
C
D
Getting Started
1
Try
1, 2, 3, 4. How many steps does it take? Now try 2, 4, 6, 8. What do you notice?2
What happens if all four starting numbers are the same?
3
What is the maximum number of steps you can find starting with single-digit numbers? Can you beat 7 steps?
Special Sequences
4
Try four consecutive square numbers:
1, 4, 9, 16 then 4, 9, 16, 25 then 9, 16, 25, 36. How many steps each time? What pattern do you notice?5
Try four consecutive cube numbers:
1, 8, 27, 64. How does this compare to squares?6
Try four consecutive triangular numbers:
1, 3, 6, 10 then 3, 6, 10, 15. What happens? Can you explain why?7
Try four consecutive Fibonacci numbers:
1, 1, 2, 3 then 2, 3, 5, 8 then 5, 8, 13, 21. Do you always get the same number of steps?Multiplying Patterns
8
Start with a number and keep doubling:
3, 6, 12, 24. Try starting from different numbers. Does the starting number matter?9
Now try tripling:
2, 6, 18, 54. What about multiplying by 4? By 5? Does the multiplier affect the number of steps?10
Try a geometric sequence with ratio ½ (switch to decimals):
8, 4, 2, 1. What about 16, 8, 4, 2?Rules and Formulas
11
Use the rule ×2 then +1: start with 1 to get
1, 3, 7, 15. Try other “multiply and add” rules. Do they all take the same number of steps?12
Use a linear rule: try
2, 5, 8, 11 (add 3 each time). Now try 3, 7, 11, 15 (add 4). Does the common difference matter?13
What happens with arithmetic sequences in general? If your numbers go up by a constant amount, can you predict the number of steps?
Hint: look at what the first layer of differences always gives you.
Deeper Questions
14
If you multiply all four starting numbers by the same amount, does it change the number of steps? Try
1, 3, 5, 9 then 10, 30, 50, 90.15
If you add the same number to all four, does it change the number of steps? Try
1, 5, 2, 8 then 101, 105, 102, 108.16
Does the order matter? Try
3, 7, 2, 9 and 7, 3, 9, 2. Are rearrangements always the same?17
Switch to fractions. Try
1/2, 1/3, 1/4, 1/5. Do fractions behave differently from whole numbers?18
Can you find four numbers where Diffy reaches zero in exactly 1 step? Exactly 2? Exactly 3? What’s the pattern?
19
Why does Diffy always reach zero? Can you convince yourself — or better yet, prove — that this process must always terminate?
Hint: what happens to the largest number at each step?
How to Play Diffy
The Idea
Start with four numbers at the corners of a square. Find the positive difference between each pair of adjacent numbers. These four differences form a new, smaller square inside the first. Repeat until you reach 0, 0, 0, 0.
How It Works
1. Enter four numbers in the corner boxes.
2. Press Start.
3. Click each ? button on the edges to reveal the difference between those two corners.
4. Once all four differences are revealed, the next inner square appears.
5. Keep going until all four corners are zero.
2. Press Start.
3. Click each ? button on the edges to reveal the difference between those two corners.
4. Once all four differences are revealed, the next inner square appears.
5. Keep going until all four corners are zero.
Example
Start with 8, 3, 5, 2.
Differences: |8−3| = 5, |3−5| = 2, |5−2| = 3, |2−8| = 6.
New square: 5, 2, 3, 6. Keep going!
Differences: |8−3| = 5, |3−5| = 2, |5−2| = 3, |2−8| = 6.
New square: 5, 2, 3, 6. Keep going!
The Challenge
Diffy always reaches zero — but some starting numbers take many more steps than others. Can you find four numbers that take the most steps? Try whole numbers, decimals, or fractions.
Number Types
Whole — integers, including negatives (e.g. 12, −3, 0).
Decimal — numbers with decimal points (e.g. 3.5, −0.25).
Fraction — enter as 3/4 or mixed numbers like 1 3/4.
Decimal — numbers with decimal points (e.g. 3.5, −0.25).
Fraction — enter as 3/4 or mixed numbers like 1 3/4.
Try These Starters
Can you find starting numbers that take the most steps?