Speechless mathematical proofs.
$begingroup$
Do you have proofs without word?
Your proofs are not necessary has zero word, you may add a bit explanations.
As an example, I has a "Speechless proof" for
$$frac{1}{4}+frac{1}{4^2}+frac{1}{4^3}+...=frac{1}{3}$$
I welcome all aspects of mathematical proofs. Thank you.
recreational-mathematics
asked Nov 5 '12 at 9:02
A. ChuA. Chu
7,01593284
$endgroup$
|
show 2 more comments
$begingroup$
Do you have proofs without word?
Your proofs are not necessary has zero word, you may add a bit explanations.
As an example, I has a "Speechless proof" for
$$frac{1}{4}+frac{1}{4^2}+frac{1}{4^3}+...=frac{1}{3}$$
I welcome all aspects of mathematical proofs. Thank you.
recreational-mathematics
asked Nov 5 '12 at 9:02
A. ChuA. Chu
7,01593284
$endgroup$
12
$begingroup$
These are usually called proofs without words. MathOverflow has a nice list of them.
$endgroup$
– Rahul
Nov 5 '12 at 9:05
$begingroup$
Please feel please to post your proof even it's similar to those in MathsOverFlow, because we are here in MathStackExchange.
$endgroup$
– A. Chu
Nov 5 '12 at 9:58
1
$begingroup$
For this question, what's really important are imaginative answers.
$endgroup$
– A. Chu
Nov 5 '12 at 10:27
1
$begingroup$
In fact, this particular example is not obvious without some words (at least to me)!
$endgroup$
– NoChance
Nov 5 '12 at 10:40
1
$begingroup$
@NoChance, you are right. Also, there is a nice proof without words for the same result here: (mathoverflow.net/a/163807). This is in the list mentioned in the comment above by Rahul.
$endgroup$
– Cyriac Antony
Jan 17 at 5:14
|
show 2 more comments
$begingroup$
Do you have proofs without word?
Your proofs are not necessary has zero word, you may add a bit explanations.
As an example, I has a "Speechless proof" for
$$frac{1}{4}+frac{1}{4^2}+frac{1}{4^3}+...=frac{1}{3}$$
I welcome all aspects of mathematical proofs. Thank you.
recreational-mathematics
asked Nov 5 '12 at 9:02
A. ChuA. Chu
7,01593284
$endgroup$
Do you have proofs without word?
Your proofs are not necessary has zero word, you may add a bit explanations.
As an example, I has a "Speechless proof" for
$$frac{1}{4}+frac{1}{4^2}+frac{1}{4^3}+...=frac{1}{3}$$
I welcome all aspects of mathematical proofs. Thank you.
recreational-mathematics
recreational-mathematics
asked Nov 5 '12 at 9:02
A. ChuA. Chu
7,01593284
asked Nov 5 '12 at 9:02
A. ChuA. Chu
7,01593284
asked Nov 5 '12 at 9:02
A. ChuA. Chu
7,01593284
asked Nov 5 '12 at 9:02
A. ChuA. Chu
7,01593284
asked Nov 5 '12 at 9:02
A. ChuA. Chu
7,01593284
7,01593284
12
$begingroup$
These are usually called proofs without words. MathOverflow has a nice list of them.
$endgroup$
– Rahul
Nov 5 '12 at 9:05
$begingroup$
Please feel please to post your proof even it's similar to those in MathsOverFlow, because we are here in MathStackExchange.
$endgroup$
– A. Chu
Nov 5 '12 at 9:58
1
$begingroup$
For this question, what's really important are imaginative answers.
$endgroup$
– A. Chu
Nov 5 '12 at 10:27
1
$begingroup$
In fact, this particular example is not obvious without some words (at least to me)!
$endgroup$
– NoChance
Nov 5 '12 at 10:40
1
$begingroup$
@NoChance, you are right. Also, there is a nice proof without words for the same result here: (mathoverflow.net/a/163807). This is in the list mentioned in the comment above by Rahul.
$endgroup$
– Cyriac Antony
Jan 17 at 5:14
|
show 2 more comments
12
$begingroup$
These are usually called proofs without words. MathOverflow has a nice list of them.
$endgroup$
– Rahul
Nov 5 '12 at 9:05
$begingroup$
Please feel please to post your proof even it's similar to those in MathsOverFlow, because we are here in MathStackExchange.
$endgroup$
– A. Chu
Nov 5 '12 at 9:58
1
$begingroup$
For this question, what's really important are imaginative answers.
$endgroup$
– A. Chu
Nov 5 '12 at 10:27
1
$begingroup$
In fact, this particular example is not obvious without some words (at least to me)!
$endgroup$
– NoChance
Nov 5 '12 at 10:40
1
$begingroup$
@NoChance, you are right. Also, there is a nice proof without words for the same result here: (mathoverflow.net/a/163807). This is in the list mentioned in the comment above by Rahul.
$endgroup$
– Cyriac Antony
Jan 17 at 5:14
12
12
$begingroup$
These are usually called proofs without words. MathOverflow has a nice list of them.
$endgroup$
– Rahul
Nov 5 '12 at 9:05
$begingroup$
These are usually called proofs without words. MathOverflow has a nice list of them.
$endgroup$
– Rahul
Nov 5 '12 at 9:05
$begingroup$
Please feel please to post your proof even it's similar to those in MathsOverFlow, because we are here in MathStackExchange.
$endgroup$
– A. Chu
Nov 5 '12 at 9:58
$begingroup$
Please feel please to post your proof even it's similar to those in MathsOverFlow, because we are here in MathStackExchange.
$endgroup$
– A. Chu
Nov 5 '12 at 9:58
1
1
$begingroup$
For this question, what's really important are imaginative answers.
$endgroup$
– A. Chu
Nov 5 '12 at 10:27
$begingroup$
For this question, what's really important are imaginative answers.
$endgroup$
– A. Chu
Nov 5 '12 at 10:27
1
1
$begingroup$
In fact, this particular example is not obvious without some words (at least to me)!
$endgroup$
– NoChance
Nov 5 '12 at 10:40
$begingroup$
In fact, this particular example is not obvious without some words (at least to me)!
$endgroup$
– NoChance
Nov 5 '12 at 10:40
1
1
$begingroup$
@NoChance, you are right. Also, there is a nice proof without words for the same result here: (mathoverflow.net/a/163807). This is in the list mentioned in the comment above by Rahul.
$endgroup$
– Cyriac Antony
Jan 17 at 5:14
$begingroup$
@NoChance, you are right. Also, there is a nice proof without words for the same result here: (mathoverflow.net/a/163807). This is in the list mentioned in the comment above by Rahul.
$endgroup$
– Cyriac Antony
Jan 17 at 5:14
|
show 2 more comments
5 Answers
5
active
oldest
votes
$begingroup$
Part of the proof in my blog post is done with my own ad-hoc diagrams..
The question is: How many "tours" (paths that visit every single square exactly once) are there in a 4x10^12 grid under the condition the tour must start in the top left square and finish in the bottom left square. (Credit to the guys at projecteuler.net for thinking up another great problem)
If we let T(n) be the formula for the number of tours in a 4xn grid, we need to find T(10^12). One approach is to find a recurrence relation. A trick is to realize there are only two possible ending columns. Try follow my working if you can, sorry It's messy :)
answered Nov 5 '12 at 9:54
robert kingrobert king
1597
$endgroup$
add a comment |
$begingroup$
The best one I have ever seen is to prove $$1 + 2 + 3 + cdots + n = dfrac{n(n+1)}2$$
$endgroup$
1
$begingroup$
Should we just copy the MO answers here - or link to them from here?
$endgroup$
– Old John
Nov 5 '12 at 9:50
2
$begingroup$
Could somebody give me the "with words" version of this proof?
$endgroup$
– littleO
Nov 5 '12 at 10:05
3
$begingroup$
@littleO There's a bijection in there. Look carefully. Perhaps it would be more helpful if we wrote $frac{n(n+1)}{2}$ as $binom{n+1}{2}$
$endgroup$
– EuYu
Nov 5 '12 at 10:09
$begingroup$
Ohh, I see, crazy. Thanks for the hint.
$endgroup$
– littleO
Nov 5 '12 at 12:00
$begingroup$
@OldJohn I don't know. I anyway made my post CW when I posted it.
$endgroup$
– user17762
Nov 5 '12 at 21:48
add a comment |
$begingroup$
Found this great one surfing the web recently.
$$
displaystyle 1/2+1/4+1/8+1/16+ldots =1
$$
edited Dec 23 '18 at 20:28
Glorfindel
3,41981830
answered Nov 5 '12 at 9:49
cyclochaoticcyclochaotic
85411429
$endgroup$
add a comment |
$begingroup$
Reciprocals of squares converge.
answered Nov 5 '12 at 10:16
EuYuEuYu
30.5k754102
$endgroup$
add a comment |
$begingroup$
Here's one showing the area of a disk is $pi R^2$.
answered Apr 16 '18 at 16:42
CharlesCharles
1527
$endgroup$
add a comment |
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5 Answers
5
active
oldest
votes
5 Answers
5
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Part of the proof in my blog post is done with my own ad-hoc diagrams..
The question is: How many "tours" (paths that visit every single square exactly once) are there in a 4x10^12 grid under the condition the tour must start in the top left square and finish in the bottom left square. (Credit to the guys at projecteuler.net for thinking up another great problem)
If we let T(n) be the formula for the number of tours in a 4xn grid, we need to find T(10^12). One approach is to find a recurrence relation. A trick is to realize there are only two possible ending columns. Try follow my working if you can, sorry It's messy :)
answered Nov 5 '12 at 9:54
robert kingrobert king
1597
$endgroup$
add a comment |
$begingroup$
Part of the proof in my blog post is done with my own ad-hoc diagrams..
The question is: How many "tours" (paths that visit every single square exactly once) are there in a 4x10^12 grid under the condition the tour must start in the top left square and finish in the bottom left square. (Credit to the guys at projecteuler.net for thinking up another great problem)
If we let T(n) be the formula for the number of tours in a 4xn grid, we need to find T(10^12). One approach is to find a recurrence relation. A trick is to realize there are only two possible ending columns. Try follow my working if you can, sorry It's messy :)
answered Nov 5 '12 at 9:54
robert kingrobert king
1597
$endgroup$
add a comment |
$begingroup$
Part of the proof in my blog post is done with my own ad-hoc diagrams..
The question is: How many "tours" (paths that visit every single square exactly once) are there in a 4x10^12 grid under the condition the tour must start in the top left square and finish in the bottom left square. (Credit to the guys at projecteuler.net for thinking up another great problem)
If we let T(n) be the formula for the number of tours in a 4xn grid, we need to find T(10^12). One approach is to find a recurrence relation. A trick is to realize there are only two possible ending columns. Try follow my working if you can, sorry It's messy :)
answered Nov 5 '12 at 9:54
robert kingrobert king
1597
$endgroup$
Part of the proof in my blog post is done with my own ad-hoc diagrams..
The question is: How many "tours" (paths that visit every single square exactly once) are there in a 4x10^12 grid under the condition the tour must start in the top left square and finish in the bottom left square. (Credit to the guys at projecteuler.net for thinking up another great problem)
If we let T(n) be the formula for the number of tours in a 4xn grid, we need to find T(10^12). One approach is to find a recurrence relation. A trick is to realize there are only two possible ending columns. Try follow my working if you can, sorry It's messy :)
answered Nov 5 '12 at 9:54
robert kingrobert king
1597
answered Nov 5 '12 at 9:54
robert kingrobert king
1597
answered Nov 5 '12 at 9:54
robert kingrobert king
1597
answered Nov 5 '12 at 9:54
robert kingrobert king
1597
1597
add a comment |
add a comment |
$begingroup$
The best one I have ever seen is to prove $$1 + 2 + 3 + cdots + n = dfrac{n(n+1)}2$$
$endgroup$
1
$begingroup$
Should we just copy the MO answers here - or link to them from here?
$endgroup$
– Old John
Nov 5 '12 at 9:50
2
$begingroup$
Could somebody give me the "with words" version of this proof?
$endgroup$
– littleO
Nov 5 '12 at 10:05
3
$begingroup$
@littleO There's a bijection in there. Look carefully. Perhaps it would be more helpful if we wrote $frac{n(n+1)}{2}$ as $binom{n+1}{2}$
$endgroup$
– EuYu
Nov 5 '12 at 10:09
$begingroup$
Ohh, I see, crazy. Thanks for the hint.
$endgroup$
– littleO
Nov 5 '12 at 12:00
$begingroup$
@OldJohn I don't know. I anyway made my post CW when I posted it.
$endgroup$
– user17762
Nov 5 '12 at 21:48
add a comment |
$begingroup$
The best one I have ever seen is to prove $$1 + 2 + 3 + cdots + n = dfrac{n(n+1)}2$$
$endgroup$
1
$begingroup$
Should we just copy the MO answers here - or link to them from here?
$endgroup$
– Old John
Nov 5 '12 at 9:50
2
$begingroup$
Could somebody give me the "with words" version of this proof?
$endgroup$
– littleO
Nov 5 '12 at 10:05
3
$begingroup$
@littleO There's a bijection in there. Look carefully. Perhaps it would be more helpful if we wrote $frac{n(n+1)}{2}$ as $binom{n+1}{2}$
$endgroup$
– EuYu
Nov 5 '12 at 10:09
$begingroup$
Ohh, I see, crazy. Thanks for the hint.
$endgroup$
– littleO
Nov 5 '12 at 12:00
$begingroup$
@OldJohn I don't know. I anyway made my post CW when I posted it.
$endgroup$
– user17762
Nov 5 '12 at 21:48
add a comment |
$begingroup$
The best one I have ever seen is to prove $$1 + 2 + 3 + cdots + n = dfrac{n(n+1)}2$$
$endgroup$
The best one I have ever seen is to prove $$1 + 2 + 3 + cdots + n = dfrac{n(n+1)}2$$
edited Nov 5 '12 at 9:22
community wiki
2 revs
user17762
1
$begingroup$
Should we just copy the MO answers here - or link to them from here?
$endgroup$
– Old John
Nov 5 '12 at 9:50
2
$begingroup$
Could somebody give me the "with words" version of this proof?
$endgroup$
– littleO
Nov 5 '12 at 10:05
3
$begingroup$
@littleO There's a bijection in there. Look carefully. Perhaps it would be more helpful if we wrote $frac{n(n+1)}{2}$ as $binom{n+1}{2}$
$endgroup$
– EuYu
Nov 5 '12 at 10:09
$begingroup$
Ohh, I see, crazy. Thanks for the hint.
$endgroup$
– littleO
Nov 5 '12 at 12:00
$begingroup$
@OldJohn I don't know. I anyway made my post CW when I posted it.
$endgroup$
– user17762
Nov 5 '12 at 21:48
add a comment |
1
$begingroup$
Should we just copy the MO answers here - or link to them from here?
$endgroup$
– Old John
Nov 5 '12 at 9:50
2
$begingroup$
Could somebody give me the "with words" version of this proof?
$endgroup$
– littleO
Nov 5 '12 at 10:05
3
$begingroup$
@littleO There's a bijection in there. Look carefully. Perhaps it would be more helpful if we wrote $frac{n(n+1)}{2}$ as $binom{n+1}{2}$
$endgroup$
– EuYu
Nov 5 '12 at 10:09
$begingroup$
Ohh, I see, crazy. Thanks for the hint.
$endgroup$
– littleO
Nov 5 '12 at 12:00
$begingroup$
@OldJohn I don't know. I anyway made my post CW when I posted it.
$endgroup$
– user17762
Nov 5 '12 at 21:48
1
1
$begingroup$
Should we just copy the MO answers here - or link to them from here?
$endgroup$
– Old John
Nov 5 '12 at 9:50
$begingroup$
Should we just copy the MO answers here - or link to them from here?
$endgroup$
– Old John
Nov 5 '12 at 9:50
2
2
$begingroup$
Could somebody give me the "with words" version of this proof?
$endgroup$
– littleO
Nov 5 '12 at 10:05
$begingroup$
Could somebody give me the "with words" version of this proof?
$endgroup$
– littleO
Nov 5 '12 at 10:05
3
3
$begingroup$
@littleO There's a bijection in there. Look carefully. Perhaps it would be more helpful if we wrote $frac{n(n+1)}{2}$ as $binom{n+1}{2}$
$endgroup$
– EuYu
Nov 5 '12 at 10:09
$begingroup$
@littleO There's a bijection in there. Look carefully. Perhaps it would be more helpful if we wrote $frac{n(n+1)}{2}$ as $binom{n+1}{2}$
$endgroup$
– EuYu
Nov 5 '12 at 10:09
$begingroup$
Ohh, I see, crazy. Thanks for the hint.
$endgroup$
– littleO
Nov 5 '12 at 12:00
$begingroup$
Ohh, I see, crazy. Thanks for the hint.
$endgroup$
– littleO
Nov 5 '12 at 12:00
$begingroup$
@OldJohn I don't know. I anyway made my post CW when I posted it.
$endgroup$
– user17762
Nov 5 '12 at 21:48
$begingroup$
@OldJohn I don't know. I anyway made my post CW when I posted it.
$endgroup$
– user17762
Nov 5 '12 at 21:48
add a comment |
$begingroup$
Found this great one surfing the web recently.
$$
displaystyle 1/2+1/4+1/8+1/16+ldots =1
$$
edited Dec 23 '18 at 20:28
Glorfindel
3,41981830
answered Nov 5 '12 at 9:49
cyclochaoticcyclochaotic
85411429
$endgroup$
add a comment |
$begingroup$
Found this great one surfing the web recently.
$$
displaystyle 1/2+1/4+1/8+1/16+ldots =1
$$
edited Dec 23 '18 at 20:28
Glorfindel
3,41981830
answered Nov 5 '12 at 9:49
cyclochaoticcyclochaotic
85411429
$endgroup$
add a comment |
$begingroup$
Found this great one surfing the web recently.
$$
displaystyle 1/2+1/4+1/8+1/16+ldots =1
$$
edited Dec 23 '18 at 20:28
Glorfindel
3,41981830
answered Nov 5 '12 at 9:49
cyclochaoticcyclochaotic
85411429
$endgroup$
Found this great one surfing the web recently.
$$
displaystyle 1/2+1/4+1/8+1/16+ldots =1
$$
edited Dec 23 '18 at 20:28
Glorfindel
3,41981830
answered Nov 5 '12 at 9:49
cyclochaoticcyclochaotic
85411429
edited Dec 23 '18 at 20:28
Glorfindel
3,41981830
edited Dec 23 '18 at 20:28
Glorfindel
3,41981830
edited Dec 23 '18 at 20:28
Glorfindel
3,41981830
3,41981830
answered Nov 5 '12 at 9:49
cyclochaoticcyclochaotic
85411429
answered Nov 5 '12 at 9:49
cyclochaoticcyclochaotic
85411429
answered Nov 5 '12 at 9:49
cyclochaoticcyclochaotic
85411429
85411429
add a comment |
add a comment |
$begingroup$
Reciprocals of squares converge.
answered Nov 5 '12 at 10:16
EuYuEuYu
30.5k754102
$endgroup$
add a comment |
$begingroup$
Reciprocals of squares converge.
answered Nov 5 '12 at 10:16
EuYuEuYu
30.5k754102
$endgroup$
add a comment |
$begingroup$
Reciprocals of squares converge.
answered Nov 5 '12 at 10:16
EuYuEuYu
30.5k754102
$endgroup$
Reciprocals of squares converge.
answered Nov 5 '12 at 10:16
EuYuEuYu
30.5k754102
answered Nov 5 '12 at 10:16
EuYuEuYu
30.5k754102
answered Nov 5 '12 at 10:16
EuYuEuYu
30.5k754102
answered Nov 5 '12 at 10:16
EuYuEuYu
30.5k754102
30.5k754102
add a comment |
add a comment |
$begingroup$
Here's one showing the area of a disk is $pi R^2$.
answered Apr 16 '18 at 16:42
CharlesCharles
1527
$endgroup$
add a comment |
$begingroup$
Here's one showing the area of a disk is $pi R^2$.
answered Apr 16 '18 at 16:42
CharlesCharles
1527
$endgroup$
add a comment |
$begingroup$
Here's one showing the area of a disk is $pi R^2$.
answered Apr 16 '18 at 16:42
CharlesCharles
1527
$endgroup$
Here's one showing the area of a disk is $pi R^2$.
answered Apr 16 '18 at 16:42
CharlesCharles
1527
answered Apr 16 '18 at 16:42
CharlesCharles
1527
answered Apr 16 '18 at 16:42
CharlesCharles
1527
answered Apr 16 '18 at 16:42
CharlesCharles
1527
1527
add a comment |
add a comment |
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12
$begingroup$
These are usually called proofs without words. MathOverflow has a nice list of them.
$endgroup$
– Rahul
Nov 5 '12 at 9:05
$begingroup$
Please feel please to post your proof even it's similar to those in MathsOverFlow, because we are here in MathStackExchange.
$endgroup$
– A. Chu
Nov 5 '12 at 9:58
1
$begingroup$
For this question, what's really important are imaginative answers.
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– A. Chu
Nov 5 '12 at 10:27
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In fact, this particular example is not obvious without some words (at least to me)!
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– NoChance
Nov 5 '12 at 10:40
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@NoChance, you are right. Also, there is a nice proof without words for the same result here: (mathoverflow.net/a/163807). This is in the list mentioned in the comment above by Rahul.
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– Cyriac Antony
Jan 17 at 5:14