- #1
Aja
- 4
- 0
How many ways can the numbers 1, 2, 5, 10, 20, 50, and 100 be combined to form the number 200?
This is a good example of the sort of problem that gives me trouble. Usually, as here, I have a few ideas but find it difficult to proceed because I get hung up on organizing the calculations to be done and on actually calculating. I lose sight of the larger problem and forget what I accomplished up till the point of my exhaustion.
Here it looks like the solution lies in grouping these numbers into all possible subgroups (.e.g.: 2, 5; 2, 5, 50, etc.).
To me the intuitively obvious way to go about this is to proceed iteratively, ordering the subgroups from those with the fewest to the most numbers (i.e., one number, two numbers, etc.). Even this simple step is a challenge. It is tedious to write out all the possibilities, and it seems presumptuous to suppose that the numbers in every subgroup sum to 200. This process as I approach it requires a lot of mental overhead, would require more than any human being could summon if this were a larger problem.
Setting these concerns aside, the second phase of the problem seems to be a matter of substituting, for each number in each subgroup, the numbers and combinations of the numbers in the original list that are their factors.
There is just too much going on here, too many things to keep in my head all at once. This sort of problem is manageable for me only if I reduce it to a simpler form with fewer numbers, less to consider. It is only easy when I consider subgroups of two numbers. When I think about three--say, one, two, and five--things become fuzzy. Here I again think iteratively:
"Three numbers, so we have three subgroups of one number: 1, 2, and 5. Err, then I seem to be overcounting." Next I try considering the largest number in the subgroup as a sort of anchor, relative to which the other two numbers accumulate with the combinations: "Okay, so how many times can one and two--well, three, go into 200--if part of the sum will be some multiple of 5? Wait, the number of ones doesn't have to equal the number of twos. I could have, say, 5 + 2(97) + 1. Wow, there are really a lot of possibilities..." And that's where I throw in the towel.
Besides help with this particular math problem, does anyone have advice for dealing with the more important mental one? For those who find this problem easy, what is solving it like for you? Do you automatically form a mental picture of its structure? Do you have a large working memory? Is there some strategy you use to keep all the details in your mind or is this just an innate ability?
Thanks. I hope somebody can shed light not just on the math here but on what trips me up. I'd like to learn math, but anytime I run into heavy calculation problems I must give up half a year or accept defeat. Very discouraging.
This is a good example of the sort of problem that gives me trouble. Usually, as here, I have a few ideas but find it difficult to proceed because I get hung up on organizing the calculations to be done and on actually calculating. I lose sight of the larger problem and forget what I accomplished up till the point of my exhaustion.
Here it looks like the solution lies in grouping these numbers into all possible subgroups (.e.g.: 2, 5; 2, 5, 50, etc.).
To me the intuitively obvious way to go about this is to proceed iteratively, ordering the subgroups from those with the fewest to the most numbers (i.e., one number, two numbers, etc.). Even this simple step is a challenge. It is tedious to write out all the possibilities, and it seems presumptuous to suppose that the numbers in every subgroup sum to 200. This process as I approach it requires a lot of mental overhead, would require more than any human being could summon if this were a larger problem.
Setting these concerns aside, the second phase of the problem seems to be a matter of substituting, for each number in each subgroup, the numbers and combinations of the numbers in the original list that are their factors.
There is just too much going on here, too many things to keep in my head all at once. This sort of problem is manageable for me only if I reduce it to a simpler form with fewer numbers, less to consider. It is only easy when I consider subgroups of two numbers. When I think about three--say, one, two, and five--things become fuzzy. Here I again think iteratively:
"Three numbers, so we have three subgroups of one number: 1, 2, and 5. Err, then I seem to be overcounting." Next I try considering the largest number in the subgroup as a sort of anchor, relative to which the other two numbers accumulate with the combinations: "Okay, so how many times can one and two--well, three, go into 200--if part of the sum will be some multiple of 5? Wait, the number of ones doesn't have to equal the number of twos. I could have, say, 5 + 2(97) + 1. Wow, there are really a lot of possibilities..." And that's where I throw in the towel.
Besides help with this particular math problem, does anyone have advice for dealing with the more important mental one? For those who find this problem easy, what is solving it like for you? Do you automatically form a mental picture of its structure? Do you have a large working memory? Is there some strategy you use to keep all the details in your mind or is this just an innate ability?
Thanks. I hope somebody can shed light not just on the math here but on what trips me up. I'd like to learn math, but anytime I run into heavy calculation problems I must give up half a year or accept defeat. Very discouraging.
