Factors & Multiples Game
Grid
Score
0
Best
—
Click any square to begin
Pick your starting number!
Starting Strategy
1
Which starting number gives you the most options for your second move? Which gives you the fewest?
Hint: think about which numbers between 1 and 100 have the most factors and multiples.
2
Is it better to start with a large number or a small number? Try starting with
1, then 50, then 97. What happens?3
What happens if you start on a prime number? What about starting on a highly composite number like
12, 24, or 60?Dangerous Numbers
4
Which numbers on the grid are “dead ends” — once you land on them, you have very few moves? Make a list.
Hint: think about primes greater than 50.
5
If you land on
97 (a prime), what are your only options? What about 89? Why are large primes so dangerous?6
Which number between 1 and 100 would be the worst possible square to land on? Can you justify your answer mathematically?
The Power of 1
7
The number
1 is a factor of every number. Does this make it a useful square to visit, or a wasteful one? Explain your reasoning.8
From
1, you can go to any unshaded number. But what can go to 1? Is there a strategic time to visit 1?Chains and Patterns
9
Try building a chain using only even numbers. How long can you make it? Now try only odd numbers. Which is easier? Why?
10
Can you build a chain that uses every multiple of 5 on the grid (5, 10, 15, …, 100)?
11
What is the longest chain you can build using only square numbers (1, 4, 9, 16, 25, 36, 49, 64, 81, 100)?
12
Try the powers of 2:
1 → 2 → 4 → 8 → 16 → 32 → 64. Does this chain work? Can you extend it?The 1–36 Grid
13
Switch to the 1–36 grid. What’s the best score you can get? Is it easier or harder than the 1–100 grid?
14
On the 1–36 grid, which starting number gives the longest possible chain? Can you shade more than 20 squares?
Deeper Thinking
15
Is it always better to go to a multiple (making the number bigger) or better to go to a factor (making it smaller)? Or does it depend?
16
Some numbers are “hubs” — connected to lots of other numbers. Which numbers between 1 and 100 are the best hubs? What do they have in common?
Hint: count both factors AND multiples within 1–100 for each number.
17
If you could extend the grid to 200, would the game be easier or harder? What about a 1–50 grid?
18
Two players play the game from the same starting number. Player A scores 15, Player B scores 28. What could Player B have done differently? Compare your chains using the Path toggle.
19
What is the theoretical maximum score? Is it possible to shade every single square on the grid? If not, what’s stopping you?
Hint: think about which numbers can ONLY be reached from one other number.
How to Play
The Goal
Shade as many squares as possible by building a chain of factors and multiples.
How It Works
1. Click any square to start your chain.
2. Then click a square that is a factor or multiple of your current number.
3. Keep going! Each move must be a factor or multiple of the last square you shaded.
4. When you get stuck, click End Turn. Your score is the number of squares in your chain.
2. Then click a square that is a factor or multiple of your current number.
3. Keep going! Each move must be a factor or multiple of the last square you shaded.
4. When you get stuck, click End Turn. Your score is the number of squares in your chain.
Example
12 → 6 (factor of 12) → 48 (multiple of 6) → 8 (factor of 48) → …
Tools
💡 Hint — highlights all available moves. Uses are counted alongside your score.
↩ Undo — go back a step if you want to try a different path.
🔢 Path — toggle to show the order you visited each square.
↩ Undo — go back a step if you want to try a different path.
🔢 Path — toggle to show the order you visited each square.
Grid Sizes
1–36 (6×6) — great for building confidence with smaller numbers.
1–50 (5×10) — a stepping stone with more options.
1–100 (10×10) — the full challenge. Rank thresholds scale to match each grid size.
1–50 (5×10) — a stepping stone with more options.
1–100 (10×10) — the full challenge. Rank thresholds scale to match each grid size.
Ranks (1–100 grid)
🔢Novice — 0+ squares
🔍Factor Finder — 5+ squares
✖️Multiple Master — 10+ squares
🔗Chain Champion — 15+ squares
➗Division Dynamo — 20+ squares
🧠Prime Strategist — 30+ squares
👑Number Theorist — 40+ squares
Keyboard Shortcuts
HToggle hints
Ctrl+ZUndo last move
EscExit full screen
End your turn?