Create your own custom ruler
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 yesterday
JonMark Perry
12.5k42461
asked yesterday
Oray
13.6k433134
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 yesterday
JonMark Perry
12.5k42461
asked yesterday
Oray
13.6k433134
2
how is 3 in one measurement?
– JonMark Perry
yesterday
1
@JonMarkPerry from 9 to 6.
– Oray
yesterday
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
yesterday
@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
yesterday
@JonMarkPerry share it please :) sounds interesting
– Oray
17 hours ago
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 yesterday
JonMark Perry
12.5k42461
asked yesterday
Oray
13.6k433134
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 yesterday
JonMark Perry
12.5k42461
asked yesterday
Oray
13.6k433134
edited yesterday
JonMark Perry
12.5k42461
edited yesterday
JonMark Perry
12.5k42461
edited yesterday
JonMark Perry
12.5k42461
12.5k42461
asked yesterday
Oray
13.6k433134
asked yesterday
Oray
13.6k433134
asked yesterday
Oray
13.6k433134
13.6k433134
2
how is 3 in one measurement?
– JonMark Perry
yesterday
1
@JonMarkPerry from 9 to 6.
– Oray
yesterday
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
yesterday
@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
yesterday
@JonMarkPerry share it please :) sounds interesting
– Oray
17 hours ago
add a comment |Â
2
how is 3 in one measurement?
– JonMark Perry
yesterday
1
@JonMarkPerry from 9 to 6.
– Oray
yesterday
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
yesterday
@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
yesterday
@JonMarkPerry share it please :) sounds interesting
– Oray
17 hours ago
2
2
how is 3 in one measurement?
– JonMark Perry
yesterday
how is 3 in one measurement?
– JonMark Perry
yesterday
1
1
@JonMarkPerry from 9 to 6.
– Oray
yesterday
@JonMarkPerry from 9 to 6.
– Oray
yesterday
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
yesterday
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
yesterday
@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
yesterday
@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
yesterday
@JonMarkPerry share it please :) sounds interesting
– Oray
17 hours ago
@JonMarkPerry share it please :) sounds interesting
– Oray
17 hours ago
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 yesterday
Jaap Scherphuis
11.9k12155
1
to be honest, i did not know the answer! thanks :)
– Oray
yesterday
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 yesterday
Jaap Scherphuis
11.9k12155
1
to be honest, i did not know the answer! thanks :)
– Oray
yesterday
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 yesterday
Jaap Scherphuis
11.9k12155
1
to be honest, i did not know the answer! thanks :)
– Oray
yesterday
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 yesterday
Jaap Scherphuis
11.9k12155
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 yesterday
Jaap Scherphuis
11.9k12155
answered yesterday
Jaap Scherphuis
11.9k12155
answered yesterday
Jaap Scherphuis
11.9k12155
answered yesterday
Jaap Scherphuis
11.9k12155
11.9k12155
1
to be honest, i did not know the answer! thanks :)
– Oray
yesterday
add a comment |Â
1
to be honest, i did not know the answer! thanks :)
– Oray
yesterday
1
1
to be honest, i did not know the answer! thanks :)
– Oray
yesterday
to be honest, i did not know the answer! thanks :)
– Oray
yesterday
add a comment |Â
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2
how is 3 in one measurement?
– JonMark Perry
yesterday
1
@JonMarkPerry from 9 to 6.
– Oray
yesterday
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
yesterday
@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
yesterday
@JonMarkPerry share it please :) sounds interesting
– Oray
17 hours ago