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Do you have any clue how the processes of folding paper and saving are connected? Think about it before reading further: what happens when you fold a piece of paper and what happens when you save money in a savings or investment account?
Take an ordinary piece of paper of the standard size 8.5"x11" (approximately). How many times can you fold this paper in half? Think about it before trying to fold. Can you do so 10 times? Write down your guess on the paper.
Let's now do the folding: fold the paper in the longer direction in half and crease it. How many layers of paper are in the result? Now fold your result again and record the number of layers in a table like the following.
| # of folds | # of layers |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | |
| . | |
| . | |
| . |
So, how many folds could you produce? If you are like most people, you could likely only fold it 5 or 6 times. Does that match your result? Here is the expanded table, with two more columns. The last column assumes that all of the creases are smooth and flat.
| # of folds | # of layers | # of layers as powers of 2 | how thick? |
|---|---|---|---|
| 0 | 1 | 20 | |
| 1 | 2 | 21 | |
| 2 | 4 | 22 | |
| 3 | 8 | 23 | |
| 4 | 16 | 24 | |
| 5 | 32 | 25 | ~0.13" |
| 6 | 64 | 26 | ~0.25" |
| 7 | 128 | 27 | ~0.5" |
| 8 | 256 | 28 | ~1" |
| 9 | 512 | 29 | ~2" |
| 10 | 1028 | 210 | ~4" |
Hopefully, you can see that as each fold occurs, the number of layers doubles. Thus, the the number of layers is growing exponentially with a base of 2.
We are going to expand the last question by continuing to do more folds. Yes, I know that you had to quit, but now we are going to imagine that we have no constraints on our strength or reach. In other words, we are going to imagine that our arms can stretch as long as needed to hold the stack of paper that is growing taller and that we have the strength to always do the next fold. We also assume that as we fold, we are able to get the paper folded "smoothly", so that it is flat as in the first few folds. The new question is the following: How many folds does it take so that the stack of folded paper would reach the moon? Think carefully about this question and what we have done so far.
Here is a quick followup question to the last question. I already gave you the answer for how many folds it would take for your folded piece of paper to reach the moon (provided that we approach this in an imaginative fashion as described above). Now I ask you how many folds it will take you to get to the moon and then back to the earth?
The title of this thought had two (key) words: Folding and Saving. This section may not seem to be related to either. Please be patient.
The Wheat and Chessboard Problem has many variations, but there is a common theme for all of them. Here is my version: Once upon a time, there was a king who needed help solving some problem or issue. There was a peasant in the king's kingdom who came up with a solution to the king's task. The king was so pleased with this peasant that he offered great riches to the peasant. The peasant declined the offer, but the king insisted on some payment and pushed the peasant to spell out his request. Finally, since the peasant played chess, he requested 1 grain of wheat for the first square on the chessboard, 2 grains on the next day on the second square, 4 grains of wheat on the third day on the third square and so on. Note that this sequence 1, 2, 4, 8, 16, 32, 64, 128 is the same as in the paper folding problem. So, what we learned from that problem applies (in some fashion) to this situation. If you wish to read more about this particular problem, check out the Wikipedia article.
What do you think of this payment? Who got the better deal? The king or the peasant? Think about it. Who and why?
Let's simply consider how many grains were placed in the first row (of this 8x8 chessboard). Therefore, we need to add up
1 + 2 + 4 + 8 + 16 + 32 + 64 + 128
[Feel free to skip some of these details, but at least try to give it a read first. It is good for your brain to tackle new things.] While it is fairly easy to add up these 8 numbers, and you might even be able to do so in your head, it will be "easier" if we change how these are represented. The eight numbers above can be written exponentially as:
20 + 21 + 22 + 23 + 24 + 25 + 26 + 27
We now want to add up these 8 numbers. First [A], we assign N to equal the sum, then [B] we multiply that equation by 2 on both sides (so, on the right, we just increase the exponent by one), and then [C] we subtract the first equation from the second equation. (Note that many pieces on the right side cancel, leaving only the first term from the top row and the last term in the second row.) [Again, try to read this.]
[A]: N = 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27
[B]: 2N = 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28
[C]: N = 28 - 20
Note that 28 - 20 = 28 -1 = 255. Next, adding up all 64 squares (so all 8 rows) and using the process above, the sum is N = 265 - 1. Note that 265=36,893,488,147,419,103,232. Therefore, the king agreed to give the peasant 36,893,488,147,419,103,231 kernels of wheat, and he, initially, thought he was getting a good deal. However, decades (or perhaps centuries) of world wheat production would not suffice to provide the gift. This is a HUGE number.
First a recap: In the folding problem, we saw that after each fold, we doubled the number of layers of the folded paper, and so the number of layers were always powers of 2 (1=2^0, 2, 4, 8, 16, 32, and so on). In the chessboard problem, each of these numbers showed up as how many kernels of wheat the peasant got on that day. We added them to find the total number of kernels, but it does not make sense to add them up in the folding problem, because the last number always records the number of layers at that time. Now we are going back to the folding problem and connecting it to saving.
Just as in the folding problem, recall that we had to use our imagination about being able to reach long distances with our arms and have extra strength to be able fold and so on, we too have to have some imagination in the initial saving problem. We are going to imagine that you have found a very generous bank (perhaps your parent), or a foolish bank, neither of which is generally true. When one puts money in a bank, the bank usually pays the one who deposits the money some interest for the use of the bank loaning out your money to people who want to buy a home, car, or other things. Of course, the bank always pays you significantly less money when they borrow your money than when they charge the ones who want to buy a home or car or something else. As of the day I write this (2026-07-07), a typical bank downtown where you live likely pays you somewhere between 0.01% and 0.10%, the national average for savings accounts is about 0.38%, and the rate for high-yield savings account at online banks might be from 3.40% to 3.60%. (I use Ally.com, which now pays 3.4%. I derive no kickback in mentioning this.) Each of these are annual rates (though often compounded more frequently - but we are not going to worry about that now). So, when you get paid 0.01% (assuming compounded once per year for each year going forward), $100 will earn 1 penny at the end of the year, while the rate 0.38% will earn almost 4 cents, and 3.4% will earn $3.40. Which would you prefer?
Now we are going to use this generous (or foolish) bank that is going to pay at the rate of 100%. So, if you put in $100 at 100% interest, at the end of the year (using simple interest, not compounded), your bank account grows to $200 since your interest is 100 X 100% = 100 x 1 = 100. Then, when you add your interest to your original amount, you get $200. Then the next year, it will double again and become $400 and the next year $800, then $1600 and so on. These numbers remind me that I forgot to tell you the other property about this bank: it only allows you to deposit $1, since it is, after all, paying 100% interest. So, at the end of the first year, you have $2, then $4, then $8, then $16 and so on. These numbers are the same numbers that appeared in the folding problem, and we recall that in that situation, the number of folds grew quite large, eventually. Here too, the bank account will grow quite large, but only one doubling per year.
This is going to give a brief example of how compounding plays a role. We continue to assume, for this example, that we still get 100% interest. Recall that $100 grew into $200 after a year, when the interest is only paid at the end of the year. Let's consider an example where the interest is compounded quarterly, or four times per year. Suppose we deposit our $100 in our bank (which actually only accepts a deposit of $1, but this is for illustrative purposes only) on Jan. 1. After three months, the bank is going to pay us interest. Since the rate is 100% but the first three months only covers one fourth of the year, we only get one fourth of the interest rate. So the interest is then $100 x 25%, which is $25. Therefore, on April 1, we will have $100 + $25 = $125. Again, three months later on June 30, we accrue our interest of 25%, but this time on our current balance, $125: $125 x 25% = $31.25. So our balance on July 1 is $125 + $31.25 = $156.25. The important thing to observe here is that the $31.25 is made up of the $25, which is the interest on the original $100, but also the $6.25 which is the interest on the first interest payement of $25. (Do you know the value of 252?) We now go another three months and on Sept. 30, we find our new interest amount to be $156.25 x 25% = $39.0625, yielding a new balance of $195.3125. Again, notice that our interest amount grew since it was interest on the original $100, plus the interest on the first interest payment and again on the second one. We do this one more time and our final interest payment is $195.3125 x 25% = $48.828125, with a new balance of $244.140625. Note that our last interest payment is almost twice that of the first one. Furthermore, we did much better than just compounding once at the end of the year, which yielded $200, compared to our $244.140625.
Why did we do so well? The biggest factor is that the interest rate is so high: 100% per year. No one pays that. The other factor is the compounding, which means collecting interest on the interest (as opposed to simple interest). A couple of observations. First, if we actually used the bank that I had described that restricted the deposit to be $1, that would have grown into $2.44140625, still better than 2. We would do even better if we compounded every month (assume 30 days), or compounded every day (both of these methods are done by some banks). While no banks do this, you could compound every hour, or minute, or second. Would this grow wildly? No. In fact if you could compound every millisecond, this fictitious bank would turn $1 into about $2.718281... Do you know this number? It is not π.
Now we are going to come back down to earth and consider a more realistic example from a bank that pays 3.4% and compare it to one that pays 0.01%. Since we are going to use a realistic bank, let's also assume that we have a realistic amount to deposit, say $1000 (which may or may not be realistic for you, depending your current financial status, but if you follow what I am talking about here, it can soon become a realistic amount). You may wish to go to https://www.calculator.net/interest-calculator.html, which is an online calculator for doing these kinds of calculations. In this calculator, if you put in $1000 for the initial investment, enter 0 for the annual contribution, change the interest rate to 3.4, change the compounding to monthly (common for banks), and change the investment length to 10 years. Now hit calculate: you should get $1404.27. Now change only the interest rate field to .01. You should see $1001.00. Which bank are you going to use? Note also that in this case, the 3% inflation great devalues your earnings and ending balance. Moral: find an online bank that pays an interest rate close to (but better to be above) the rate of inflation.
One of the ways to greatly increase the return of your money is to regularly commit to regularly add more to your account. Let's go back to the 3.4% example from above and start with all the other values as given earlier, but change the monthly contribution to $100/month. So each month, not only do we get our interest, but we also add more money. This should result in an ending balance of $15,713.13. Our interest was much greater than the previous example because we regularly added $100 more each month.
Note that this title changed from Saving to Investing. A word of explanation is needed. BEFORE you start investing in mutual funds or exchange traded funds (ETFs) that contain stocks of companies, one should first fill up their savings account so that you can handle emergencies, medical issues, accidents, being laid off, and more. A rule of thumb is to have 3 to 6 (but 9 is better) months of your take-home pay (after taxes and other deductions have been paid). Once that is done, then you might consider investing. A former colleague and I have been giving sessions to upper-class college students for almost two decades and some more details can found at the website that we prepared. This site also (briefly) talks about pre-tax, Roth, and taxable accounts. Here is just one example for what one might consider for investing. Let's image that you are 22 and that you are going to work until you are 62, so for 40 years. You start out with no investments (so put 0 into the first box), but decide to contribute $250/month and assume that the rate of return is 7% (which is realistic for a long-term stock investment). This example should return an ending balance of well over half a million ($660K).
We have seen that savings (and investments) grow as a function of their interest rate (and a number of other factors). For example, putting your money into a bank paying 3.4% earns considerably more than 0.01%. However, 7% or 10% does even much better (though more risky - over the short term). A simple way to compare rates of return is to use the rule of 72. In the first examples, we saw the power of doubling (leading to exponential growth), so an important question is to know long it will take for your funds/money to double in value at various rates of return. Suppose that you have $50,000 in either a bank or an investment (e.g., a bank or an ETF like Vanguard's VTI) and you are earning 1% per year. Now take 72/1 = 72. That $50,000 will become $100,000 (i.e., doubled) after 72 years. But if you earn 2%, it will double in 72/2 = 36 years. Or if you earn 3%, it will be doubled in 72/3 = 24 years. Note that this is simply an estimate, not exact. Consider the following table.
| annual interest rate | years for a doubling |
|---|---|
| 1 | 72 |
| 2 | 36 |
| 3 | 24 |
| 3.4 | 21.2 |
| 4 | 18 |
| 6 | 12 |
| 8 | 9 |
| 9 | 8 |
| 12 | 6 |
As alluded to above, the higher the claimed rate of return, the more risky is the investment. You can NOT get a 12% rate of return in a safe investment for a consistent window of time. It might happen, but you can't count on it (despite what the huckster selling you something says). Don't get greedy. Just use this rule as a simple guideline. If you have more questions about saving, investing and related topics, I encourage you to first review the pages at my college's website that I mentioned earlier. You can also contact me with questions, after you have read that site.
Published 2026-07-07.
If you find any errors in the text, please let me know. Thanks. Contact me with errors or comments using hibbardac@gmail.com. [Back to the top] [About the author, Al]
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