Worked Numbers
The 100 Envelope Challenge: How Much It Totals, and a Pace You Can Keep
The 100 envelope challenge is simple: number 100 envelopes from $1 to $100, pick one at random, and put that many dollars in it. When every envelope is full, you have saved $5,050.
Why it’s $5,050
Pair the envelopes from the outside in: $1 with $100, $2 with $99, $3 with $98. Every pair adds up to $101, and there are 50 pairs. The average envelope is $50.50.
The total can come as a surprise if you think of the challenge as “a few dollars at a time”. Some envelopes are a few dollars. The average one is $50.50.
What it costs per week
How fast you go is up to you. Here is what each pace means, by envelopes per week:
| Per week | Weeks | Avg week | Worst week |
|---|---|---|---|
| 1 | 100 | $50.50 | $100 |
| 2 | 50 | $101.00 | $199 |
| 3 | 34 | $148.53 | $297 |
| 4 | 25 | $202.00 | $394 |
| 7 | 15 | $336.67 | $679 |
“Worst week” is the unlucky case where you draw the biggest envelopes together. A common version - two envelopes a week - finishes in 50 weeks at an average of $101.00 a week, about $439 a month. That is a serious savings plan, not a game, and it is worth knowing before you start rather than in week ten. If the money is meant for a bill you can name in advance, a sinking fund sizes the monthly amount from the bill instead.
Making it survivable
- Draw, but allow a swap. If you draw $94 in a tight week, put it back and draw again. The total doesn’t change; only the order does.
- Go slower. One envelope a week takes 100 weeks - just under two years - at an average of $50.50 a week.
- Pick a smaller board. The 52-week challenge (week 1 is $1, week 52 is $52) totals $1,378. The 365-day penny challenge (day 1 is one cent, day 365 is $3.65) totals $667.95.
Envelopes or not
The challenge works with real envelopes and cash, or with a printed chart you color in while the money goes to a savings account. The chart version avoids keeping thousands of dollars in cash at home - and the coloring-in is most of the motivation either way.
How these were calculated
Every total on this page was added up square by square and also from the formula for the sum of 1 to n; the page only builds if they agree. The pace table is computed from the same 100 amounts.
If you’d rather color in a board than stuff envelopes, these are seven printable boards - the 100-envelope board, both 52-week versions, the penny challenge, $1,000 and $2,500 boards, and a blank one - each summed square by square:
And if you do use cash, these fold up from a single sheet, sized so a bill slides in flat, with a running ledger on the back: