stellar numbers solution

7/27/2019 Stellar Numbers Solution

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STELLAR NUMBERS SOLUTION

RICHARD CRAIG, III

ACADEMY OF RICHMOND COUNTY

[email protected]


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Stellar Numbers Solution

The first pattern shown in the stellar numbers task is called the triangular numbers. It becomes obvious

from the graphical representation that each figure adds a new row. The number of dots in the additional

row is one greater than the number in the largest row of the previous figure. That is to say that to move

from the 2nd

figure to the 3rd

, we add 3 dots. To move from the 3rd

figure to the 4th

, we add four dots. Ingeneral, to move from figure n to figure n+1, we add n+1 additional dots. A regression finds the explicit

formula to be

J$ - J

where represents the total number of dots and n represents the term number. A more approachable

equation may be

{#

$,

which means we simply multiply the current term number by the next term number and take half in

order to find the number of dots.

The following table shows the number of dots for the first four figures.

n un

1 1

2 13

3 37

4 73

To move from n=1 to n=2, 12 dots are added. From n=2 to n=3, 24 dots are added. Finally, from n=3 to

n=4, 36 dots are added. The increase in dots increase itself each time by 12 more dots. This is because

the outside star of each stellar shape has twice as many dots as the outside star of the previous stellar

shape. The outside star in S2 has 12 dots, S3s outside star has 24 and S4s outside star has 36. This

pattern is confirmed by drawing the next few stellar figures. A table of values for 1n10 is shown

below.

n un

1 1

2 13

3 37

4 73

5 121

6 181

7 253

8 337

9 433

10 541

An expression to represent the 6-stellar number at stage S7 is


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S7=1+12+24+36+48+60+72=253.

A regression easily finds a general statement for the 6-stellar number at stage Sn in terms ofn to be

J$ . J -

Let us now consider the 8-stellar numbers. The first few diagrams are shown below:

The table below counts the dots in each figure:

n Sn

1 1

2 17

3 49

4 97Note that the amount of dots in the outer layer (the amount each figure increases by) is 16 in S2 and 32

in S3 and 48 in S4. Following this pattern, we can easily deduce the amount of dots in each figure, and

the following chart shows that for 1n10.

n Sn

1 1

2 17

3 49

4 97

5 1616 241

7 337

8 449

9 577

10 721

A regression finds the formula for the sum, Sn, of the 8-stellar numbers to be


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%J$ . %J -

Following these same patters, it is easy to deduce the p-stellar numbers for other values of p. The

following table shows p={4,5,7,9,10}.

p 1 2 3 4 5 6 7 8 9 10

n Sn Sn Sn Sn Sn Sn Sn Sn Sn Sn

1 1 1 1 1 1 1 1 1 1 1

2 3 5 7 9 11 13 15 17 19 21

3 7 13 19 25 31 37 43 49 55 61

4 13 25 37 49 61 73 85 97 109 121

5 21 41 61 81 101 121 141 161 181 201

6 31 61 91 121 151 181 211 241 271 301

7 43 85 127 169 211 253 295 337 379 421

8 57 113 169 225 281 337 393 449 505 561

9 73 145 217 289 361 433 505 577 649 721

10 91 181 271 361 451 541 631 721 811 901

The following graphs show a scatterplot of n versus Sn for each value of p from 1 to 5.

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90

80

70

60

50

40

30

20

10

0

n

p=1

Scatterplot of p=1 vs n

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200

150

100

50

0

n

p=2



5/6

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300

250

200

150

100

50

0

n

p

=3


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400

300

200

100

0

n

p=4


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500

400

300

200

100

0

n

p=5


The following table shows the general statement for value of p, obtained from using a simple regression:


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p

1

2

3

4

56

7

8

9

10

In general, it is easy to see that the

It should be clear that because the

n,p .

*Note validity should be checked

Richard Craig [email protected]

Equation

n2-n+1

2n2-2n+1

3n2-3n+1

4n2-4n+1

5n2

-5n+16n

2-6n+1

7n2-7n+1

8n2-8n+1

9n2-9n+1

10n2-10n+1

equation for Sn of the p-stellar numbers is

alues of this equation represent a number of dots

and an informal justification provided. Please send

*

in a diagram, that

any comments to

stellar numbers solution

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