I need some help with this number pattern

[rfl]

The natural numbers are arranged in the following way:
2 3 4 5 6 7
13 12 11 10 9 8
14 15 16 17 18 19
25 24 23 22 21 20
In what row and column will 150 be in. What about 1001 or 5237?
Given that any natural number N>1, how can we determine wich row and column N will be in?

 

[Deleted member 4993]

The natural numbers are arranged in the following way:
2 3 4 5 6 7
13 12 11 10 9 8
14 15 16 17 18 19
25 24 23 22 21 20
In what row and column will 150 be in. What about 1001 or 5237?
Given that any natural number N>1, how can we determine wich row and column N will be in?

Hint: in each row there are 6 numbers - starting from 2

Largest number in row 1 = 6 = (2-1) + 1*6

Largest number in row 2 = 13 = (2-1) + 2*6

Largest number in row 3 = 19 = (2-1) + 3*6

This way you can find which row does a number belong. And then continue...

Please show us your work, indicating exactly where you are stuck - so that we know where to begin to help you.

 

[TchrWill]

The natural numbers are arranged in the following way:
ROW
\/
.........1...2...3...4...6...7 <Col

1........2...3...4...5...6...7

2.......13..12..11..10...9...8

3.......14..15..16..16..18..19

4.......25..24..23..22..21..20

In what row and column will 150 be in. What about 1001 or 5237?
Given that any natural number N>1, how can we determine wich row and column N will be in?

N = 6n + 1 = 150

n = 6

Even rows increase by when moving right in the row.
Odd rows increase by when moving left in the row.

Since n is even, 150 must be in column 1.

N = 6n + 1 = 1001

n = 166.6666...

Since the integer part of n is 166,or even, 1001 must be in row 167 in column 6 (.66666(6)) = 5

Try 5237 yourself..

 

[Loren]

I'm not seeing how Tchrwill arrived at the solution. I'm thinking that the first number in each even numbered row is represented by 6n+2 where n is the number of the even numbered row. If we develop the inequality...6n+2 < 150, then, solving for n we get n < 16 2/3. Therefore 150 is somewhere above the first number in row 16. If n=16 then the first number in row 16 is 6(16)+2 which is 98. Since 98 is the first number in row 16, 99 is in the second column and 100 is in the 3rd column. My answer would be 150 is in the 16th row and the 3rd column.
Am I not seeing something?

I am adding a second thought. My solution is assuming that the number of the row is even. If we find the number of the row to be odd, then we must start our count at the end of that row and work to the left. For instance, suppose we are looking for the location of 45.
The first number in each even numbered row can be represented by 6n+1. But in the even numbered rows, the values decrease from left to right. Therefore, our inequality is...
6n+1 > 45
6n > 44
n > 7 2/6
Since n>7 the next row number would be 8. So, 45 has to be in the 8th row. The first number in row 8 is 6(8)+1=49. So, 49 is in column 1, 48 is in column 2, 47 is in column 3, 46 is in column 4 and 47 is column 5.
45 is in row 8, column 5.

I'm sure there is a more refined way to express this mathematically. I won't be devoting any more time trying to find it.

 

[soroban]

\(\displaystyle \text{Hello, rfl!}\)

\(\displaystyle \text{I }think\text{ I have a procedure . . .}\)

\(\displaystyle \text{The natural numbers are arranged in the following way:}\)

. . . \(\displaystyle \begin{array}{cccccc} 2&3&4&5&6&7 \\ 13&12&11&10&9&8 \\ 14&15&16&17&18&19 \\ 25&24&23&22&21&20 \end{array}\)

\(\displaystyle \text{In what row and column will 150 be in? }\:\text{What about 1001 or 537 ?}\)

\(\displaystyle \text{Given any integer }N>1\text{, how can we determine which row and column }N\text{ will be in?}\)

\(\displaystyle \text{Given }N\text{: subtract 2, divide by 6 and round down to the nearest integer, then add 1.}\)
. . . . \(\displaystyle \text{This gives the Row Number, }\:R \;=\;\left[\frac{N-2}{6}\right] + 1\)
\(\displaystyle \text{Note whether }R\text{ is odd or even.}\)



\(\displaystyle \text{To determine the column number }C,\:\text{ divide }N-2\text{ by 6, and note the remainder, }r.\)

. . . . \(\displaystyle \text{If }R\text{ is odd, the column number is: }\:C \:=\:r + 1\)

. . . . \(\displaystyle \text{If }R\text{ is even, the column number is: }\:C \:=\:6-r\)


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


\(\displaystyle \text{Example: }\:N = 17\)

. . . . \(\displaystyle \left[\frac{15}{6}\right] + 1 \quad\Rightarrow\quad R \:=\:3\;\text{(odd)}\)

. . . . \(\displaystyle 15 \div 6 \:=\:2,\;r=3 \quad\Rightarrow\quad C \:=\:3+1 \:=\:4\)

\(\displaystyle \text{Therefore, 17 is at }(3,4).\)



\(\displaystyle \text{Example: }\:N = 21\)

. . . . \(\displaystyle \left[\frac{19}{6}\right] + 1 \quad\Rightarrow\quad R \:=\:4\:\text{(even)}\)

. . . . \(\displaystyle 19 \div 6 \:=\:3,\:r=1 \quad\Rightarrow\quad C \:=\:6-1 \:=\:5\)

\(\displaystyle \text{Therefore, 21 is at }(4,5).\)



\(\displaystyle \text{Example: }\:N = 537\)

. . . . \(\displaystyle \left[\frac{535}{6}\right]+1 \quad\Rightarrow\quad R \:=\:90\:\text{(even)}\)

. . . . \(\displaystyle 535 \div 6 \:=\:89,\:r = 1 \quad\Rightarrow\quad C \:=\:6-1 \:=\:5\)

\(\displaystyle \text{Therefore, 537 is at }(90,5).\)



\(\displaystyle \text{Example: }\:N = 1001\)

. . . . \(\displaystyle \left[\frac{999}{6}\right] + 1 \quad\Rightarrow\quad R \:=\:167\:\text{(odd)}\)

. . . . \(\displaystyle 999 \div 6 \:=\:166,\:r = 3 \quad\Rightarrow\quad C \:=\:3+1 \:=\:4\)

\(\displaystyle \text{Therefore, 1001 is at }(167,4).\)

 

[Denis]

Mine is slightly different, but agrees with Soroban's:

R = ceiling[(n - 1)/6]

If R even, C = 6R - n + 2, else C = n - 6R + 5

1001: R = 167, C = 4
5237: R = 873, C = 4

 

[TchrWill]

I'm not seeing how Tchrwill arrived at the solution. I'm thinking that the first number in each even numbered row is represented by 6n+2 where n is the number of the even numbered row. If we develop the inequality...6n+2 < 150, then, solving for n we get n < 16 2/3. Therefore 150 is somewhere above the first number in row 16. If n=16 then the first number in row 16 is 6(16)+2 which is 98. Since 98 is the first number in row 16, 99 is in the second column and 100 is in the 3rd column. My answer would be 150 is in the 16th row and the 3rd column.
Am I not seeing something?

I am adding a second thought. My solution is assuming that the number of the row is even. If we find the number of the row to be odd, then we must start our count at the end of that row and work to the left. For instance, suppose we are looking for the location of 45.
The first number in each even numbered row can be represented by 6n+1. But in the even numbered rows, the values decrease from left to right. Therefore, our inequality is...
6n+1 > 45
6n > 44
n > 7 2/6
Since n>7 the next row number would be 8. So, 45 has to be in the 8th row. The first number in row 8 is 6(8)+1=49. So, 49 is in column 1, 48 is in column 2, 47 is in column 3, 46 is in column 4 and 47 is column 5.
45 is in row 8, column 5.

I'm sure there is a more refined way to express this mathematically. I won't be devoting any more time trying to find it.

Some clarification.

I did error in transcribing my notes to the original post. I think the following will clarify things.

The natural numbers are arranged in the following way:
ROW
\/
.........1...2...3...4...6...7 <Col

1........2...3...4...5...6...7

2.......13..12..11..10...9...8

3.......14..15..16..16..18..19

4.......25..24..23..22..21..20

In what row and column will 150 be in. What about 1001 or 5237?
Given that any natural number N>1, how can we determine wich row and column N will be in?

The odd numbered rows increase by 1 when moving right in a row from column 1 to 6.
The even numbered rows decrease by 1 when moving left in a row from column 6 to 1 or increase by 1 when moving left in a row from column 6 to 1.

The first numbers in column 1 in the even numbered rows are given by Ne(1) = 6n + 1.
The numbers in column 6 in the even numbered rows are given by Ne(6) = 6n - 5.
The first numbers in column 1 in the odd numbered rows are given by No(1) = 6(n-1)2.
The first numbers in column 6 in the odd numbered rows are given by No(6) = 6n + 1.

N = 6n + 1 = 150

n = 24.83333...

This tells us that 150 is loacted in row 24+ = 25 and column 6(.833333) = 5.

N = 6n + 1 = 1001

n = 166.6666...

Since the integer part of n is 166,or even, 1001 must be in row 167 in column 6(.83333) = 5

Try 5237 yourself.

Sorry for the mixup.