Why am I encountering the binary system in this problem?

[musicgold]

Hi,
This is not homework. I need to understand why the binary number system works in the case of a particular problem.

While trying to solve the puzzle below, I set out to finding the minimum number of denominations required to create a particular amount. To create any amount up to $15, I just need to have 4 coin types: 1, 2, 4, 8. Then I realized this problem is somehow related to the binary number system or the $2^n$ system.

"A dealer has 1000 one-dollar coins and 10 bags. He has to divide the coins over the ten bags, so that he can make any number of coins simply by handing over a few bags. How must divide his money into the ten bags?"

1. Homework Statement

a. Why is this issue related to the binary system or doubling successive amounts? Why, for example, is it not related $3^n$ or $5^n$? Is it some how related to the nature of reality?

b. Why do we almost never see $4 or $8 coins or bills in any society, even though they are the building blocks (as we saw above) to create any amount?

Homework Equations

The Attempt at a Solution

Thanks

 

[Mark44]

Hi,
This is not homework. I need to understand why the binary number system works in the case of a particular problem.

While trying to solve the puzzle below, I set out to finding the minimum number of denominations required to create a particular amount. To create any amount up to $15, I just need to have 4 coin types: 1, 2, 4, 8. Then I realized this problem is somehow related to the binary number system or the $2^n$ system.

"A dealer has 1000 one-dollar coins and 10 bags. He has to divide the coins over the ten bags, so that he can make any number of coins simply by handing over a few bags. How must divide his money into the ten bags?"

1. Homework Statement

a. Why is this issue related to the binary system or doubling successive amounts? Why, for example, is it not related $3^n$ or $5^n$? Is it some how related to the nature of reality?

It has nothing to do with the nature of reality. As you mentioned, with four coin types of 1, 2, 4, and 8 units, we can make any number between 0 and 15 simply by including or not including one of the types. You never need more than one of the coin types to make up a number.

If the system were based on powers of 3 or powers of 5, you would need multiples of each of the coin types to get certain values. For example, to make 8, you would need two 3's and two 1's. With a binary system of coins all you need is one 8. With a system based on powers of 5, to get 8 you need one 5 and three 1's.

b. Why do we almost never see $4 or $8 coins or bills in any society, even though they are the building blocks (as we saw above) to create any amount?

Most likely because we have 10 fingers, with 5 on each hand. That's the most reasonable rationale for why our money system has coins that are multples of 5 and 10.

Homework Equations

The Attempt at a Solution

Thanks

 

[haruspex]

If the system were based on powers of 3 or powers of 5, you would need multiples of each of the coin types to get certain values. For example, to make 8, you would need two 3's and two 1's.

To expand on that, using base 3 you would need 2 bags of 1, 2 bags of 3, 2 bags of 9... You would run out of bags at a total of 242 coins.
The question becomes, why is this less efficient?
It's to do with choice. In the binary scheme, there is a unique way of making each sum, whereas in the ternary we often have choices: to make 4 we have to choose one of the two 1 bags and one of the two 3 bags. Given 10 bags, there are only 1024 different subsets we can form. Having different subsets representing the same total is wasteful.