1Two ways to grow a pile
You can add, or you can double
Watch the dots stack up. One way climbs in a straight ramp. The other one bends — it grows faster the bigger it gets.
Add the same step
Put the same amount on top every time. Each step is the same height, so the pile climbs in a straight line. Fair and steady.
Double it
Add a slice of whatever you already have. The more you've got, the bigger the next jump — so the pile bends upward and speeds up.
2Meet the two racers
Steady Sam vs Doubling Dot
+$1,000 every single day
Sam starts with nothing and drops $1,000 on his pile every day, like clockwork. A thousand bucks a day! After 5 days he has $5,000. He looks unbeatable.
One penny that doubles every day
Dot starts with just 1 penny. Each day her pile doubles: 1¢, 2¢, 4¢, 8¢… After 5 days she has 16 cents. Sixteen cents! It looks completely hopeless.
3The big question
Who wins on day 30? 🏁
Steady Sam adds $1,000 a day — a huge, fixed amount. Doubling Dot only has a few dollars even after two weeks. Guess first. Then you'll get the race to run yourself, all the way to the end.
Guess before you find out
It's day 30. Sam has been adding $1,000 every single day. Dot's penny has doubled every single day. Who has more money?
Your turn to run the race. The teal bar is Steady Sam, the gold bar is Doubling Dot. Drag the slider all the way to day 30 — keep your eye on the gold bar and watch the exact day it catches up, then leaves Sam in the dust. For the first two weeks the penny is so tiny you can barely see it; flip "zoom in on the penny" to spot it.
4So who's actually better?
Each one trades something
Adding a fixed amount is safe and easy to count on. If the race had ended on day 15, Sam would have won by a mile.
Once it gets rolling, growing by a slice of yourself bends upward and rockets away from any straight line.
Adding the same thing grows in a straight line. Growing by a slice of yourself bends upward — and given enough time, the bender always beats the line, even if it starts as a single penny.
Psst, grown-ups: this is the gap between linear growth (constant addition, y = a + b·x) and exponential growth (constant ratio, y = a·rⁿ). A penny doubling daily has r = 2; by day 30 it's 2²⁹ pennies ≈ $5.37 million, versus Sam's 30 × $1,000 = $30,000. Any exponential with r > 1 overtakes any straight line eventually, no matter how small its start or how steep the line — here the crossover is day 23. The same math powers compound interest, populations, and viral spread; real systems only stay exponential until a limit (resources, saturation) bends the curve into an S.