(023) (024) (033) (034) (044)
(111) (112) (113) (114) (122)
(123) (124) (133) (134) (144)
(222) (223) (224) (233) (234)
(244) (333) (334) (344) (444)
Using numbers 0, 1, 2, 3, 4 and 5
(000) (001) (002) (003) (004) (005)
(011) (012) (013) (014) (015) (022)
(023) (024) (025) (033) (034) (035)
(044) (045) (055) (111) (112) (113)
(114) (115) (122) (123) (124) (125)
(133) (134) (135) (144) (145) (155)
(222) (223) (224) (225) (233) (234)
(235) (244) (245) (255) (333) (334)
(335) (344) (345) (355) (444) (445)
(455) (555)
Using numbers 0, 1, 2, 3, 4, 5 and 6
(000) (001) (002) (003) (004) (005)
(006) (011) (012) (013) (014) (015)
(016) (022) (023) (024) (025) (026)
(033) (034) (035) (036) (044) (045)
(046) (055) (056) (066) (111) (112)
(113) (114) (115) (116) (122) (123)
(124) (125) (126) (133) (134) (135)
(136) (144) (145) (146) (155) (156)
(166) (222) (223) (224) (225) (226)
(233) (234) (235) (236) (244) (245)
(246) (255) (256) (266) (333) (334)
(335) (336) (344) (345) (346) (355)
(356) (366) (444) (445) (446) (455)
(456) (466) (555) (556) (566) (666)
RESULTS TABLE
Largest Number
1 2 3 4 5 6
+1 +1 +1 +1 +1
Linear Equation
FORMULA
f (n) =an + b
f (n) = 1n + 0
Sum of all numbers
6 30 90 210 420 756
+24 +60 +120 +210 +336
+36 +60 +90 +126
+24 +30 +36
+6 +6
Quartic Equation
f (n) =an4+bn3+cn2+dn+e
n=1
a + b + c + d + e = 6
n=2
16a + 8b + 4c + 2d + e = 30
n=3
81a + 27b + 9c + 3d + e = 90
n=4
256a + 64b + 16c +4d + e = 210
n=5
625a + 125b + 25c + 5d + e = 420
Equation 5 – 4
369a + 61b + 9c + d = 210
Equation 4 – 3
175a + 37b + 7c + d = 120
Equation 3 – 2
65a + 19b + 5c + d = 60
Equation 2 – 1
15a+ 7b + 3c + d = 24
Equation 6 – 7
194a + 24b + 2c = 90
Equation 7 – 8
110a + 18b + 2c = 60
Equation 8 – 9
50a + 12b + 2c =36
Equation 10 – 11
84a + 6b = 30
Equation 11 – 12
60a + 6b = 24
Equation 13 – 14
24a = 6
= 6÷24
= 0.25
a = 0.25
Equation 13
84a + 6b = 30
21 + 6b =30
6b = 30 – 21
= 9
b = 9 ÷ 6
b = 1.5
Equation 10
194a + 24b + 2c = 90
- + 36 + 2c = 90
84.5 + 2c = 90
2c = 90 – 84.5
c = 5.5 /2
c = 2.75
Equation 9
15a+ 7b + 3c + d = 24
3.75 + 10.5 + 8.25 + d = 24
22.5 + d = 24
d = 24 – 22.5
d = 1.5
Equation 1
a + b + c + d + e = 6
0.25 + 1.5 + 2.75 + 1.5 + e = 6
6 + e = 6
e = 6-6
e = 0
CHECK
Equation 1
a + b + c + d + e = 6
0.25 + 1.5 + 2.75 + 1.5 + 0 = 6
FORMULA
f (n) =an4+bn3+cn2+dn+e
f (n) = 0.25n4 + 1.5n3 + 2.75n2 + 1.5n + 0
This formula shows the sum of all the numbers. To find the sum of the numbers using the numbers 0, 1, 2 and 3 this is how it works:
f (n) = 0.25n4 + 1.5n3 + 2.75n2 + 1.5n + 0
f (4) = 64 + 96 + 44 + 6
f (4) = 210
The formula was correct because it gave the right answer.
Number of Cards
1 4 10 20 35 56 84
+3 +6 +10 +15 +21 +28
+3 +4 +5 +6 +7
+1 +1 +1 +1
Cubic Equation
f (n) =an3+bn2+cn+d
n=1
a + b + c + d = 4
n=2
8a + 4b + 2c + d = 10
n=3
27a + 9b + 3c + d = 20
n=4
64a + 16b + 4c + d = 35
Equation 4 – 3
37a + 7b + c = 15
Equation 3 – 2
19a + 5b + c = 10
Equation 2 – 1
7a + 3b + c = 6
Equation 5 – 6
18a + 2b = 5
Equation 6 – 7
12a + 2b = 4
Equation 8 – 9
6a = 1
a = 1/6
Equation 8
18a + 2b = 5
3 + 2b = 5
2b = 5 – 3
b = 2/2
b = 1
Equation 7
7a + 3b + c = 6
1.166666667 + 3 + c = 6
4.166666667+ c = 6
c = 6 - 4.166666667
c = 11/6
Equation 1
a + b + c + d = 4
1/6 + 1 + 11/6 + d = 4
3 + d = 4
d = 4- 3
d = 1
CHECK
Equation 1
a + b + c + d = 4
1/6+1+11/6+1= 4
FORMULA
f (n) =an3+bn2+cn+d
f (n) =1/6n3+1n2+11/6n+1
This formula shows the sum of all the numbers. To find the sum of the numbers using the numbers 0, 1, 2 and 3 this is how it works:
f (n) =1/6n3+1n2+11/6n+1
f (4) =1/6(4)3+16+11/6(4) +1
f (4) =35
The formula was correct because it gave the right answer.
f (n) =1/6n3+1n2+11/6n+1 (Number of Cards)
f (n) = 0.25n4 + 1.5n3 + 2.75n2 + 1.5n + 0 (Sum of all numbers)
Factorizing Sum of All Numbers Formula
0.25n (n3+6n2+11n+6)
Factorizing Number of Cards
1/6 (n3+n2+11n+6)
Comparing 2 formulas
0.25n (n3+6n2+11n+6)
1/6 (n3+n2+11n+6)
Both of the formulas are very similar there are only a few differences between them for example in the first formula the letter n is outside the formula whereas in the formula there is no letter n situated outside the bracket. In the first formula there is a number 6 in front of the n2 whereas there is no number 6 in front of the n2 in the second formula.
Connection between Sum Of Numbers And Number Of Cards And The Largest Number
To find the connection I divided 6 by 4 the answer was 1.5, to check if this was the connection I tested it on the second pair of cards I multiplied 10 by 1.5 the answer was 15 then I multiplied it by the largest number which was 2 the answer was 30 this was correct. The connection between the sum of numbers and number of cards is that you multiply the number of cards by 1.5 then you multiply it by the largest number you should get the sum of numbers as your final answer.
CONCLUSION
The main aim of the investigation was to find the relationship between the cubic formula and the quartic formula, I have found a few difference between the two formulas I done this by factorizing both formulas then looking for the differences. I was able to solve the patterns in the equations. I was also able to find the connection between the sum of cards, the largest number and the number of cards. I was also able to draw graphs of the equations; sadly the graphs are not complete as you could see if I had continued to draw the graph another curve would have appeared. If I were to do this investigation again I would perhaps use different numbers.
AHSAN AHMED Candidate Number 9065