4. Next we recorded our results in a results table.
4. We then found formulae for them.
Results:
All results were recorded in a table, so if a pattern were to occur, it would be easier to spot.
Next we tried to see if there was a pattern like square numbers, cube numbers, triangular numbers, odd numbers or even numbers.
We found out the following:
0 Face: 0,0,1,8,27 (Tasneem with the help of the revision guide found out that these numbers were cube numbers).
1 Face:0, 0,6,24,54 (Asma found out these numbers were from the 6 times table).
2 Face: 0,0,12,24,36 (Alliyah and Aysha.k found these numbers were in the 12 times table).
3 Face: 0,8,8,8,8 (We all found that no matter what the size of the cube, 3 faces painted always stayed 8).
Formulae:
First we thought the following were the correct formulae but then we found out that they were all incorrect.
O faces painted:-
Each time you times the number by itself 3 times, so we thought the formula would be: n3
1 face painted:-
Each number is from the 6 times tables, so we thought the formula would be:
n+6
2 faces painted:-
Each time you add 12, so we thought the formula would be: n+12
3 faces painted:-
Each time it is the same, so we thought the formula would be: n+0
By substituting different values for n, we found these formulae were wrong, as it was not returning the values we had in our result’s table.
What we did next
Before we found out the 0 face painted column were cubes numbers and with the help of Miss Elahi found the formula is (n-2)3 not sure how she got the –2 part but when we tested we found it was correct.
With help of Miss Elahi the formula for 1 face painted is 6n2-24n+24. Still not sure how Miss got it but when we tested it was correct.
With Aysha.K sister’s (Badeeia) help and revision guide we found out the formula for 2 faces painted. First we found the difference it was 12. You add 12 each time so we thought it was 12n. When we tested it we found it was wrong. It has to be 12n – something. For the first one you had to take away 24 to get back to the original answer. We tried 12n-24 for the rest and it was right. Therefore the formula is 12n-24.
Before we found out the 3 face painted column was always 8 and found out there is no formula n is always = to 8 so n=8. This is because there are always 4 corners in any sized cube.
Final Formulae
0 Face Painted: (n-2)3
1 Face Painted: 6n2-24n+24
2 Face Painted: 12n-24
3 Face Painted: Always 8 n=8
Conclusion
We were able to find all the formulas. We found out that the 0 face painted column were
all cubes numbers The 1 face painted column are in the 6 times table. The 2 faces painted column are in the 12 times table. Also we found out the 3 faces painted never changes.
Evaluation
To make this investigation you should test more different sized cubes.