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Create your own custom ruler
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There is a blank ruler of $X$ units. You are going to set $9$ marks on this ruler so that you will be able to measure all integer values from $1$ to $X$ units with only one measurement.
What is the maximum value $X$ can take?
For example: if this problem was asked for $3$ marks, the answer would be $9$ by marking $1$,$2$ and $6$ units on the ruler.
Source: 2006 Puzzleup
mathematics logical-deduction optimization
edited Aug 18 at 8:24
JonMark Perry
13.3k42666
asked Aug 18 at 7:28
Oray
14.1k435139
add a comment |Â
up vote
8
down vote
favorite
There is a blank ruler of $X$ units. You are going to set $9$ marks on this ruler so that you will be able to measure all integer values from $1$ to $X$ units with only one measurement.
What is the maximum value $X$ can take?
For example: if this problem was asked for $3$ marks, the answer would be $9$ by marking $1$,$2$ and $6$ units on the ruler.
Source: 2006 Puzzleup
mathematics logical-deduction optimization
edited Aug 18 at 8:24
JonMark Perry
13.3k42666
asked Aug 18 at 7:28
Oray
14.1k435139
2
how is 3 in one measurement?
– JonMark Perry
Aug 18 at 8:18
1
@JonMarkPerry from 9 to 6.
– Oray
Aug 18 at 8:19
Yes, the end points $0$ and $X$ are essentially also marks, not included in the count of the additional marks that you need to make. The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for. It looks possible to find the answer by hand, but I'm not so sure how you could prove that $X$ is maximal without computer assistance.
– Jaap Scherphuis
Aug 18 at 8:32
@JaapScherphuis "The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for." find it then and post it here as an answer. I believe it is also a solution to find an answer via wiki. I didnt say you should not use a computer to solve it by the way.
– Oray
Aug 18 at 8:34
@JonMarkPerry share it please :) sounds interesting
– Oray
Aug 19 at 5:36
add a comment |Â
up vote
8
down vote
favorite
up vote
8
down vote
favorite
There is a blank ruler of $X$ units. You are going to set $9$ marks on this ruler so that you will be able to measure all integer values from $1$ to $X$ units with only one measurement.
What is the maximum value $X$ can take?
For example: if this problem was asked for $3$ marks, the answer would be $9$ by marking $1$,$2$ and $6$ units on the ruler.
Source: 2006 Puzzleup
mathematics logical-deduction optimization
edited Aug 18 at 8:24
JonMark Perry
13.3k42666
asked Aug 18 at 7:28
Oray
14.1k435139
There is a blank ruler of $X$ units. You are going to set $9$ marks on this ruler so that you will be able to measure all integer values from $1$ to $X$ units with only one measurement.
What is the maximum value $X$ can take?
For example: if this problem was asked for $3$ marks, the answer would be $9$ by marking $1$,$2$ and $6$ units on the ruler.
Source: 2006 Puzzleup
mathematics logical-deduction optimization
edited Aug 18 at 8:24
JonMark Perry
13.3k42666
asked Aug 18 at 7:28
Oray
14.1k435139
edited Aug 18 at 8:24
JonMark Perry
13.3k42666
edited Aug 18 at 8:24
JonMark Perry
13.3k42666
edited Aug 18 at 8:24
JonMark Perry
13.3k42666
13.3k42666
asked Aug 18 at 7:28
Oray
14.1k435139
asked Aug 18 at 7:28
Oray
14.1k435139
asked Aug 18 at 7:28
Oray
14.1k435139
14.1k435139
2
how is 3 in one measurement?
– JonMark Perry
Aug 18 at 8:18
1
@JonMarkPerry from 9 to 6.
– Oray
Aug 18 at 8:19
Yes, the end points $0$ and $X$ are essentially also marks, not included in the count of the additional marks that you need to make. The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for. It looks possible to find the answer by hand, but I'm not so sure how you could prove that $X$ is maximal without computer assistance.
– Jaap Scherphuis
Aug 18 at 8:32
@JaapScherphuis "The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for." find it then and post it here as an answer. I believe it is also a solution to find an answer via wiki. I didnt say you should not use a computer to solve it by the way.
– Oray
Aug 18 at 8:34
@JonMarkPerry share it please :) sounds interesting
– Oray
Aug 19 at 5:36
add a comment |Â
2
how is 3 in one measurement?
– JonMark Perry
Aug 18 at 8:18
1
@JonMarkPerry from 9 to 6.
– Oray
Aug 18 at 8:19
Yes, the end points $0$ and $X$ are essentially also marks, not included in the count of the additional marks that you need to make. The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for. It looks possible to find the answer by hand, but I'm not so sure how you could prove that $X$ is maximal without computer assistance.
– Jaap Scherphuis
Aug 18 at 8:32
@JaapScherphuis "The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for." find it then and post it here as an answer. I believe it is also a solution to find an answer via wiki. I didnt say you should not use a computer to solve it by the way.
– Oray
Aug 18 at 8:34
@JonMarkPerry share it please :) sounds interesting
– Oray
Aug 19 at 5:36
2
2
how is 3 in one measurement?
– JonMark Perry
Aug 18 at 8:18
how is 3 in one measurement?
– JonMark Perry
Aug 18 at 8:18
1
1
@JonMarkPerry from 9 to 6.
– Oray
Aug 18 at 8:19
@JonMarkPerry from 9 to 6.
– Oray
Aug 18 at 8:19
Yes, the end points $0$ and $X$ are essentially also marks, not included in the count of the additional marks that you need to make. The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for. It looks possible to find the answer by hand, but I'm not so sure how you could prove that $X$ is maximal without computer assistance.
– Jaap Scherphuis
Aug 18 at 8:32
Yes, the end points $0$ and $X$ are essentially also marks, not included in the count of the additional marks that you need to make. The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for. It looks possible to find the answer by hand, but I'm not so sure how you could prove that $X$ is maximal without computer assistance.
– Jaap Scherphuis
Aug 18 at 8:32
@JaapScherphuis "The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for." find it then and post it here as an answer. I believe it is also a solution to find an answer via wiki. I didnt say you should not use a computer to solve it by the way.
– Oray
Aug 18 at 8:34
@JaapScherphuis "The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for." find it then and post it here as an answer. I believe it is also a solution to find an answer via wiki. I didnt say you should not use a computer to solve it by the way.
– Oray
Aug 18 at 8:34
@JonMarkPerry share it please :) sounds interesting
– Oray
Aug 19 at 5:36
@JonMarkPerry share it please :) sounds interesting
– Oray
Aug 19 at 5:36
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
7
down vote
accepted
A ruler with just a few marks is called a "sparse ruler". Such a ruler of length $X$ that can measure all lengths from $1$ to $X$ is called a "complete sparse ruler". It is called optimal if it can't be done with fewer marks and there is no complete ruler with the same number of marks which is longer.
The Wikipedia page for Sparse Ruler lists optimal sparse rulers, and it shows that the one for 11 marks (including the end points of the ruler) is unique:
It has length $X=43$ and the marks are at $0, 1, 3, 6, 13, 20, 27, 34, 38, 42, 43$
II.I..I......I......I......I......I...I...II
01234567890123456789012345678901234567890123
The middle sections are all of length $7$, so it is fairly easy to verify that all lengths are measurable.
answered Aug 18 at 8:52
Jaap Scherphuis
12.4k12155
1
to be honest, i did not know the answer! thanks :)
– Oray
Aug 18 at 9:24
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
A ruler with just a few marks is called a "sparse ruler". Such a ruler of length $X$ that can measure all lengths from $1$ to $X$ is called a "complete sparse ruler". It is called optimal if it can't be done with fewer marks and there is no complete ruler with the same number of marks which is longer.
The Wikipedia page for Sparse Ruler lists optimal sparse rulers, and it shows that the one for 11 marks (including the end points of the ruler) is unique:
It has length $X=43$ and the marks are at $0, 1, 3, 6, 13, 20, 27, 34, 38, 42, 43$
II.I..I......I......I......I......I...I...II
01234567890123456789012345678901234567890123
The middle sections are all of length $7$, so it is fairly easy to verify that all lengths are measurable.
answered Aug 18 at 8:52
Jaap Scherphuis
12.4k12155
1
to be honest, i did not know the answer! thanks :)
– Oray
Aug 18 at 9:24
add a comment |Â
up vote
7
down vote
accepted
A ruler with just a few marks is called a "sparse ruler". Such a ruler of length $X$ that can measure all lengths from $1$ to $X$ is called a "complete sparse ruler". It is called optimal if it can't be done with fewer marks and there is no complete ruler with the same number of marks which is longer.
The Wikipedia page for Sparse Ruler lists optimal sparse rulers, and it shows that the one for 11 marks (including the end points of the ruler) is unique:
It has length $X=43$ and the marks are at $0, 1, 3, 6, 13, 20, 27, 34, 38, 42, 43$
II.I..I......I......I......I......I...I...II
01234567890123456789012345678901234567890123
The middle sections are all of length $7$, so it is fairly easy to verify that all lengths are measurable.
answered Aug 18 at 8:52
Jaap Scherphuis
12.4k12155
1
to be honest, i did not know the answer! thanks :)
– Oray
Aug 18 at 9:24
add a comment |Â
up vote
7
down vote
accepted
up vote
7
down vote
accepted
A ruler with just a few marks is called a "sparse ruler". Such a ruler of length $X$ that can measure all lengths from $1$ to $X$ is called a "complete sparse ruler". It is called optimal if it can't be done with fewer marks and there is no complete ruler with the same number of marks which is longer.
The Wikipedia page for Sparse Ruler lists optimal sparse rulers, and it shows that the one for 11 marks (including the end points of the ruler) is unique:
It has length $X=43$ and the marks are at $0, 1, 3, 6, 13, 20, 27, 34, 38, 42, 43$
II.I..I......I......I......I......I...I...II
01234567890123456789012345678901234567890123
The middle sections are all of length $7$, so it is fairly easy to verify that all lengths are measurable.
answered Aug 18 at 8:52
Jaap Scherphuis
12.4k12155
A ruler with just a few marks is called a "sparse ruler". Such a ruler of length $X$ that can measure all lengths from $1$ to $X$ is called a "complete sparse ruler". It is called optimal if it can't be done with fewer marks and there is no complete ruler with the same number of marks which is longer.
The Wikipedia page for Sparse Ruler lists optimal sparse rulers, and it shows that the one for 11 marks (including the end points of the ruler) is unique:
It has length $X=43$ and the marks are at $0, 1, 3, 6, 13, 20, 27, 34, 38, 42, 43$
II.I..I......I......I......I......I...I...II
01234567890123456789012345678901234567890123
The middle sections are all of length $7$, so it is fairly easy to verify that all lengths are measurable.
answered Aug 18 at 8:52
Jaap Scherphuis
12.4k12155
answered Aug 18 at 8:52
Jaap Scherphuis
12.4k12155
answered Aug 18 at 8:52
Jaap Scherphuis
12.4k12155
answered Aug 18 at 8:52
Jaap Scherphuis
12.4k12155
12.4k12155
1
to be honest, i did not know the answer! thanks :)
– Oray
Aug 18 at 9:24
add a comment |Â
1
to be honest, i did not know the answer! thanks :)
– Oray
Aug 18 at 9:24
1
1
to be honest, i did not know the answer! thanks :)
– Oray
Aug 18 at 9:24
to be honest, i did not know the answer! thanks :)
– Oray
Aug 18 at 9:24
add a comment |Â
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2
how is 3 in one measurement?
– JonMark Perry
Aug 18 at 8:18
1
@JonMarkPerry from 9 to 6.
– Oray
Aug 18 at 8:19
Yes, the end points $0$ and $X$ are essentially also marks, not included in the count of the additional marks that you need to make. The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for. It looks possible to find the answer by hand, but I'm not so sure how you could prove that $X$ is maximal without computer assistance.
– Jaap Scherphuis
Aug 18 at 8:32
@JaapScherphuis "The solution to this problem can easily be found on Wikipedia if you know the mathematical term to look for." find it then and post it here as an answer. I believe it is also a solution to find an answer via wiki. I didnt say you should not use a computer to solve it by the way.
– Oray
Aug 18 at 8:34
@JonMarkPerry share it please :) sounds interesting
– Oray
Aug 19 at 5:36