The Partitions of 12.
Rahul Dey 9P Version 1.2
By Rahul Dey
3. Coursework Version History
4. Question 1
5-9. Question 2
10. Question 3
Coursework Project History
Version 1.0 – Added all of the needed information into the coursework.
Version 1.1 – Neatened up layout of partitions, and elaborated briefly on some ideas
Version 1.2 – Put all of the partitions into tables to save space, and to neaten it up.
The Partitions of 12:
Using the following systematic process, the number 12 can be partitioned into the following pairs:
(NOTE: I HAVE ALSO PUT THE PRODUCT COLUMN HERE TO SHOW THE PRODUCTS OF THE TWO PARTITIONING INTEGERS WHEN MULTIPLIED)
Due to the fact that there are infinite partitions of the number 12 (and indeed any other number), I have only broken them down into halves, to show that it works with fractional numbers.
The answer to question 1 is 6x6. This is because 6x6 gives the highest product when multiplies together.
I notice that there are some interesting facts regarding the answer.
- 6 is no ordinary number, it is actually half of 12. When you multiply something by ½, you are actually also dividing it by 2. This is interesting because we are trying to find pairs and pairs mean two of something.
- 6 must multiply by itself to produce the highest product. This is also another name for squaring. Since squaring a number involves the power of 2, it also interesting.
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However, to prove any similarities in other partitioning, I will have to actually test more.
The second question involves me choosing numbers of my own. I have chosen three numbers. They are: 5, 20 and 35.
I have chosen 5 because it is a small odd number (whereas 12 was a small even number).
I have chosen 20 because it is quite a large even number.
I have chosen 35 because it is quite a large odd number.
I have chosen these numbers to see if there are any similarities to different situations.
Partitions of 5:
Again, here 2.5 has the highest product out of all of the partitions. I think that there is a pattern emerging. It seems that the halves of the numbers have the highest product. I will still investigate the other two numbers before I come to a conclusion.
Partitions of 20:
Again, I see that the middle number gives the highest product. I have now established a formula for partition into pairs.
n = (a / 2)²
'n' is the maxi-product.
'a' is the number to be partitioned.
Using this new formula, I will try and predict the maxi-product for the next number, which is 35.
n = (35 / 2)²
∴ n = 17.5²
306.25 is my prediction for the maxi-product of 35. Now, I will check if I am correct.
These tables have proved my theory, and therefore I can establish my previous formula as the formula for finding out all pairs of numbers.
Triples of the number 12
I will now split the number 12 into triples (that is 3 parts). They are all listed below:
The highest product given is here is 64, which is from 4x4x4. It is interesting that these are all the same and there are three of them and we are trying to find triples.
Therefore, with this information, I can create the formula:
n = (b ÷ 3)³
Quadruples with 12:
I shall now split the number 12 into quadruples (that is, 4 parts):
Again, I have noticed that the partition that has the highest product is always the one that has all of the partitioned integers the same.
Therefore, I can create the formula for quadruples here:
I have noticed that all of the formulae have many similarities. If x = total number to be partitioned; and p = the number of partitions necessary, I can make the following formula as a universal calculation that will provide the answer to any of these types of queries:
Question 3 says to examine any kind of number in all cases and with any number of partitions.
I have chosen a few numbers that will be listed under sub-headings.
I have chosen pi because like some other numbers, it is irrational, meaning it has no end of notation. Therefore, I will have to use a calculator to achieve the partitions of it, however, technically, none of my answers will be correct.
I will partition pi into quadruples. HOWEVER, since I cannot split pi exactly I will have to rely on the formula to help me.
Therefore, my working would be as follows:
The Number 1:
I have chosen the number one because it has no whole numbers before, therefore it will need fractional numbers. I will split it into triples. However, I will only go down to 1/3s.
The formula also works for this and still produces the same answer.