Error in solution of Peter Winkler “red and blue dice” puzzle?

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10














This question relates to the solution give in Peter Winkler's Mathematical Mind-Benders to the "Red and Blue Dice" problem appearing on page $23.$




You have two sets (one red, one blue) of $n n-$sided dice, each die labeled with the numbers from $1$ to $n.$ You roll all $2n$ dice simultaneously. Prove that there must be a nonempty subset of the red dice and a nonempty subset of the blue dice with the same sum!




I tried to prove it by induction. There must be an $n$ rolled or we can remove one die of each color and get a counterexample to the $n-1$ case. If there is only on $n$ rolled, we can remove it, and any die of the other color, and again get a counterexample. So there are at least two red $n$'s, say. But I couldn't carry the induction idea any further. I proved it up to $n=6,$ hoping to spot a pattern, but all I got was a collection of ad hoc arguments. After several days, I gave up and looked at the answer.



A solution is given on pages $33-34.$ Winkler advises proving a stronger statement.




In fact, there is a much stronger statement than the one you were asked to prove, which is nonetheless still true. Organize the red dice into a line, in any way you want, and do the same with the blue dice. Then there is a contiguous nonempty subset of each line with the same sum.



To put it more mathematically, given any two vector $langle a_1,dots,a_nrangle$ and $langle b_1,dots,b_nrangle$ in $1,dots,n^n,$ there are $jle k$ and $sle t$ such that $sum_i=j^ka_i=sum_i=s^tb_i.$



To see this, let $alpha_m$ be the sum of the first $m a_i$'s and let $beta_m$ be the sum of the first $m b_i$'s. Assume that $alpha_nlebeta_n$ (otherwise we can switch the roles of the $a$'s and $b$'s), and for each $m,$ let $m'$ be the greatest index for which $beta_m'lealpha_m.$




Winkler gives a diagram with two sample strings, lines joining $a_m$ and $b_m'$ labeled by $alpha_m-beta_m'$



enter image description here



It is apparent that the $a_i$ are the dice on top and the $b_i$ those on bottom. Note that we have $alpha_6=22, beta_6=18,$ contradicting $alpha_nlebeta_n,$ so I imagine that the latter was a typo. Also, the line labeled $3$ joining $a_3$ and $b_4$ should really end at $b_5$ and be labeled $0,$ but I guess this is just a mistake.



Anyway, Winkler says,




We always have $alpha_m-beta_m'ge0,$ and at most $n-1$ (if $alpha_m-beta_m'$ were larger than or equal to $n, m'$ would have been a larger index.)




He then goes on to observe that if any of the labels is $0$ we are done, so we have $n$ labels from $1$ to $n-1$ and two are equal. Then the sum of the intervening dice must be the same. For example, in the picture we have two lines labeled $2,$ and we have $6+5+3=3+2+3+6.$



It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong. But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?



I thought about abandoning the stronger statement, and attempting to solve the puzzle by arranging the $a_i$ in decreasing order and the $b_i$ in increasing order, but I don't see how to dispose of the case $maxa_i>minb_i.$ Winkler's argument can't be applied, and I don't see how to dispose of it otherwise.



I haven't been able to rescue this proof. Am I overlooking something? Can you solve the puzzle?



Note: Winkler say that some similar results can be found in a paper by Diaconis, Graham, and Sturmfels. I haven't tried to read the paper yet, but it looks a little heavy for the solution to a puzzle. Also, Winkler says that the source of the puzzle was David Kempe of USC, "who needed the result in a computer science paper," but gives no further reference.



P.S.



I found a list of David Kempe's publications, but I can't tell which is likely to contain a proof of the theorem.










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    10














    This question relates to the solution give in Peter Winkler's Mathematical Mind-Benders to the "Red and Blue Dice" problem appearing on page $23.$




    You have two sets (one red, one blue) of $n n-$sided dice, each die labeled with the numbers from $1$ to $n.$ You roll all $2n$ dice simultaneously. Prove that there must be a nonempty subset of the red dice and a nonempty subset of the blue dice with the same sum!




    I tried to prove it by induction. There must be an $n$ rolled or we can remove one die of each color and get a counterexample to the $n-1$ case. If there is only on $n$ rolled, we can remove it, and any die of the other color, and again get a counterexample. So there are at least two red $n$'s, say. But I couldn't carry the induction idea any further. I proved it up to $n=6,$ hoping to spot a pattern, but all I got was a collection of ad hoc arguments. After several days, I gave up and looked at the answer.



    A solution is given on pages $33-34.$ Winkler advises proving a stronger statement.




    In fact, there is a much stronger statement than the one you were asked to prove, which is nonetheless still true. Organize the red dice into a line, in any way you want, and do the same with the blue dice. Then there is a contiguous nonempty subset of each line with the same sum.



    To put it more mathematically, given any two vector $langle a_1,dots,a_nrangle$ and $langle b_1,dots,b_nrangle$ in $1,dots,n^n,$ there are $jle k$ and $sle t$ such that $sum_i=j^ka_i=sum_i=s^tb_i.$



    To see this, let $alpha_m$ be the sum of the first $m a_i$'s and let $beta_m$ be the sum of the first $m b_i$'s. Assume that $alpha_nlebeta_n$ (otherwise we can switch the roles of the $a$'s and $b$'s), and for each $m,$ let $m'$ be the greatest index for which $beta_m'lealpha_m.$




    Winkler gives a diagram with two sample strings, lines joining $a_m$ and $b_m'$ labeled by $alpha_m-beta_m'$



    enter image description here



    It is apparent that the $a_i$ are the dice on top and the $b_i$ those on bottom. Note that we have $alpha_6=22, beta_6=18,$ contradicting $alpha_nlebeta_n,$ so I imagine that the latter was a typo. Also, the line labeled $3$ joining $a_3$ and $b_4$ should really end at $b_5$ and be labeled $0,$ but I guess this is just a mistake.



    Anyway, Winkler says,




    We always have $alpha_m-beta_m'ge0,$ and at most $n-1$ (if $alpha_m-beta_m'$ were larger than or equal to $n, m'$ would have been a larger index.)




    He then goes on to observe that if any of the labels is $0$ we are done, so we have $n$ labels from $1$ to $n-1$ and two are equal. Then the sum of the intervening dice must be the same. For example, in the picture we have two lines labeled $2,$ and we have $6+5+3=3+2+3+6.$



    It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong. But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?



    I thought about abandoning the stronger statement, and attempting to solve the puzzle by arranging the $a_i$ in decreasing order and the $b_i$ in increasing order, but I don't see how to dispose of the case $maxa_i>minb_i.$ Winkler's argument can't be applied, and I don't see how to dispose of it otherwise.



    I haven't been able to rescue this proof. Am I overlooking something? Can you solve the puzzle?



    Note: Winkler say that some similar results can be found in a paper by Diaconis, Graham, and Sturmfels. I haven't tried to read the paper yet, but it looks a little heavy for the solution to a puzzle. Also, Winkler says that the source of the puzzle was David Kempe of USC, "who needed the result in a computer science paper," but gives no further reference.



    P.S.



    I found a list of David Kempe's publications, but I can't tell which is likely to contain a proof of the theorem.










    share|cite|improve this question


























      10












      10








      10


      2





      This question relates to the solution give in Peter Winkler's Mathematical Mind-Benders to the "Red and Blue Dice" problem appearing on page $23.$




      You have two sets (one red, one blue) of $n n-$sided dice, each die labeled with the numbers from $1$ to $n.$ You roll all $2n$ dice simultaneously. Prove that there must be a nonempty subset of the red dice and a nonempty subset of the blue dice with the same sum!




      I tried to prove it by induction. There must be an $n$ rolled or we can remove one die of each color and get a counterexample to the $n-1$ case. If there is only on $n$ rolled, we can remove it, and any die of the other color, and again get a counterexample. So there are at least two red $n$'s, say. But I couldn't carry the induction idea any further. I proved it up to $n=6,$ hoping to spot a pattern, but all I got was a collection of ad hoc arguments. After several days, I gave up and looked at the answer.



      A solution is given on pages $33-34.$ Winkler advises proving a stronger statement.




      In fact, there is a much stronger statement than the one you were asked to prove, which is nonetheless still true. Organize the red dice into a line, in any way you want, and do the same with the blue dice. Then there is a contiguous nonempty subset of each line with the same sum.



      To put it more mathematically, given any two vector $langle a_1,dots,a_nrangle$ and $langle b_1,dots,b_nrangle$ in $1,dots,n^n,$ there are $jle k$ and $sle t$ such that $sum_i=j^ka_i=sum_i=s^tb_i.$



      To see this, let $alpha_m$ be the sum of the first $m a_i$'s and let $beta_m$ be the sum of the first $m b_i$'s. Assume that $alpha_nlebeta_n$ (otherwise we can switch the roles of the $a$'s and $b$'s), and for each $m,$ let $m'$ be the greatest index for which $beta_m'lealpha_m.$




      Winkler gives a diagram with two sample strings, lines joining $a_m$ and $b_m'$ labeled by $alpha_m-beta_m'$



      enter image description here



      It is apparent that the $a_i$ are the dice on top and the $b_i$ those on bottom. Note that we have $alpha_6=22, beta_6=18,$ contradicting $alpha_nlebeta_n,$ so I imagine that the latter was a typo. Also, the line labeled $3$ joining $a_3$ and $b_4$ should really end at $b_5$ and be labeled $0,$ but I guess this is just a mistake.



      Anyway, Winkler says,




      We always have $alpha_m-beta_m'ge0,$ and at most $n-1$ (if $alpha_m-beta_m'$ were larger than or equal to $n, m'$ would have been a larger index.)




      He then goes on to observe that if any of the labels is $0$ we are done, so we have $n$ labels from $1$ to $n-1$ and two are equal. Then the sum of the intervening dice must be the same. For example, in the picture we have two lines labeled $2,$ and we have $6+5+3=3+2+3+6.$



      It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong. But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?



      I thought about abandoning the stronger statement, and attempting to solve the puzzle by arranging the $a_i$ in decreasing order and the $b_i$ in increasing order, but I don't see how to dispose of the case $maxa_i>minb_i.$ Winkler's argument can't be applied, and I don't see how to dispose of it otherwise.



      I haven't been able to rescue this proof. Am I overlooking something? Can you solve the puzzle?



      Note: Winkler say that some similar results can be found in a paper by Diaconis, Graham, and Sturmfels. I haven't tried to read the paper yet, but it looks a little heavy for the solution to a puzzle. Also, Winkler says that the source of the puzzle was David Kempe of USC, "who needed the result in a computer science paper," but gives no further reference.



      P.S.



      I found a list of David Kempe's publications, but I can't tell which is likely to contain a proof of the theorem.










      share|cite|improve this question















      This question relates to the solution give in Peter Winkler's Mathematical Mind-Benders to the "Red and Blue Dice" problem appearing on page $23.$




      You have two sets (one red, one blue) of $n n-$sided dice, each die labeled with the numbers from $1$ to $n.$ You roll all $2n$ dice simultaneously. Prove that there must be a nonempty subset of the red dice and a nonempty subset of the blue dice with the same sum!




      I tried to prove it by induction. There must be an $n$ rolled or we can remove one die of each color and get a counterexample to the $n-1$ case. If there is only on $n$ rolled, we can remove it, and any die of the other color, and again get a counterexample. So there are at least two red $n$'s, say. But I couldn't carry the induction idea any further. I proved it up to $n=6,$ hoping to spot a pattern, but all I got was a collection of ad hoc arguments. After several days, I gave up and looked at the answer.



      A solution is given on pages $33-34.$ Winkler advises proving a stronger statement.




      In fact, there is a much stronger statement than the one you were asked to prove, which is nonetheless still true. Organize the red dice into a line, in any way you want, and do the same with the blue dice. Then there is a contiguous nonempty subset of each line with the same sum.



      To put it more mathematically, given any two vector $langle a_1,dots,a_nrangle$ and $langle b_1,dots,b_nrangle$ in $1,dots,n^n,$ there are $jle k$ and $sle t$ such that $sum_i=j^ka_i=sum_i=s^tb_i.$



      To see this, let $alpha_m$ be the sum of the first $m a_i$'s and let $beta_m$ be the sum of the first $m b_i$'s. Assume that $alpha_nlebeta_n$ (otherwise we can switch the roles of the $a$'s and $b$'s), and for each $m,$ let $m'$ be the greatest index for which $beta_m'lealpha_m.$




      Winkler gives a diagram with two sample strings, lines joining $a_m$ and $b_m'$ labeled by $alpha_m-beta_m'$



      enter image description here



      It is apparent that the $a_i$ are the dice on top and the $b_i$ those on bottom. Note that we have $alpha_6=22, beta_6=18,$ contradicting $alpha_nlebeta_n,$ so I imagine that the latter was a typo. Also, the line labeled $3$ joining $a_3$ and $b_4$ should really end at $b_5$ and be labeled $0,$ but I guess this is just a mistake.



      Anyway, Winkler says,




      We always have $alpha_m-beta_m'ge0,$ and at most $n-1$ (if $alpha_m-beta_m'$ were larger than or equal to $n, m'$ would have been a larger index.)




      He then goes on to observe that if any of the labels is $0$ we are done, so we have $n$ labels from $1$ to $n-1$ and two are equal. Then the sum of the intervening dice must be the same. For example, in the picture we have two lines labeled $2,$ and we have $6+5+3=3+2+3+6.$



      It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong. But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?



      I thought about abandoning the stronger statement, and attempting to solve the puzzle by arranging the $a_i$ in decreasing order and the $b_i$ in increasing order, but I don't see how to dispose of the case $maxa_i>minb_i.$ Winkler's argument can't be applied, and I don't see how to dispose of it otherwise.



      I haven't been able to rescue this proof. Am I overlooking something? Can you solve the puzzle?



      Note: Winkler say that some similar results can be found in a paper by Diaconis, Graham, and Sturmfels. I haven't tried to read the paper yet, but it looks a little heavy for the solution to a puzzle. Also, Winkler says that the source of the puzzle was David Kempe of USC, "who needed the result in a computer science paper," but gives no further reference.



      P.S.



      I found a list of David Kempe's publications, but I can't tell which is likely to contain a proof of the theorem.







      combinatorics puzzle integer-partitions






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      edited Dec 11 at 19:01

























      asked Dec 11 at 16:17









      saulspatz

      13.8k21328




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          2 Answers
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          7














          The figure is mistaken, but the proof is not, after a small clarification. Ignore the figure.



          Winkler intended to assume $alpha_nle beta_n$. Furthermore, he intended $0$ to be an allowable index when choosing $m'$, where $beta_0=0$. This ensures $m'$ exists. For each $mge 1$, we have $alpha_m ge beta_0$. This implies that $i:0le ile n,alpha_mge beta_i$ is nonempty; possibly it only contains $i=0$. We then let $m'$ be the largest element of this set.



          By definition, $alpha_m-beta_m'ge 0$. If $alpha_m-beta_m'ge n$, then it must be that $m'<n$, because $alpha_mle alpha_n le beta_n$. We can then consider $beta_m'+1$, and would have $beta_m'+1=beta_m'+((m'+1)^sttext dice)le beta_m'+nle alpha_m$, contradicting the maximality of $m'$. Therefore you have $0le alpha_m-beta_m'le n-1$ and the rest of the proof follows.






          share|cite|improve this answer




















          • Thanks. I tried setting $1'=0,$ but somehow I couldn't make it work.
            – saulspatz
            Dec 11 at 18:49


















          3















          It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong.




          Yes, take $alpha_nleq beta_n.$




          But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?




          Set $1'=0.$ When you take the intervening dice between two different "$m$"'s the lower "$m$" is excluded, so it's fine to use zero here. The higher "$m$" is included, but that's ok: if $alpha_m-beta_m'=alpha_M-beta_M'=c$ with $m<M$ then necessarily $M'>0$ because $beta_M'>beta_m'.$






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            2 Answers
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            2 Answers
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            active

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            7














            The figure is mistaken, but the proof is not, after a small clarification. Ignore the figure.



            Winkler intended to assume $alpha_nle beta_n$. Furthermore, he intended $0$ to be an allowable index when choosing $m'$, where $beta_0=0$. This ensures $m'$ exists. For each $mge 1$, we have $alpha_m ge beta_0$. This implies that $i:0le ile n,alpha_mge beta_i$ is nonempty; possibly it only contains $i=0$. We then let $m'$ be the largest element of this set.



            By definition, $alpha_m-beta_m'ge 0$. If $alpha_m-beta_m'ge n$, then it must be that $m'<n$, because $alpha_mle alpha_n le beta_n$. We can then consider $beta_m'+1$, and would have $beta_m'+1=beta_m'+((m'+1)^sttext dice)le beta_m'+nle alpha_m$, contradicting the maximality of $m'$. Therefore you have $0le alpha_m-beta_m'le n-1$ and the rest of the proof follows.






            share|cite|improve this answer




















            • Thanks. I tried setting $1'=0,$ but somehow I couldn't make it work.
              – saulspatz
              Dec 11 at 18:49















            7














            The figure is mistaken, but the proof is not, after a small clarification. Ignore the figure.



            Winkler intended to assume $alpha_nle beta_n$. Furthermore, he intended $0$ to be an allowable index when choosing $m'$, where $beta_0=0$. This ensures $m'$ exists. For each $mge 1$, we have $alpha_m ge beta_0$. This implies that $i:0le ile n,alpha_mge beta_i$ is nonempty; possibly it only contains $i=0$. We then let $m'$ be the largest element of this set.



            By definition, $alpha_m-beta_m'ge 0$. If $alpha_m-beta_m'ge n$, then it must be that $m'<n$, because $alpha_mle alpha_n le beta_n$. We can then consider $beta_m'+1$, and would have $beta_m'+1=beta_m'+((m'+1)^sttext dice)le beta_m'+nle alpha_m$, contradicting the maximality of $m'$. Therefore you have $0le alpha_m-beta_m'le n-1$ and the rest of the proof follows.






            share|cite|improve this answer




















            • Thanks. I tried setting $1'=0,$ but somehow I couldn't make it work.
              – saulspatz
              Dec 11 at 18:49













            7












            7








            7






            The figure is mistaken, but the proof is not, after a small clarification. Ignore the figure.



            Winkler intended to assume $alpha_nle beta_n$. Furthermore, he intended $0$ to be an allowable index when choosing $m'$, where $beta_0=0$. This ensures $m'$ exists. For each $mge 1$, we have $alpha_m ge beta_0$. This implies that $i:0le ile n,alpha_mge beta_i$ is nonempty; possibly it only contains $i=0$. We then let $m'$ be the largest element of this set.



            By definition, $alpha_m-beta_m'ge 0$. If $alpha_m-beta_m'ge n$, then it must be that $m'<n$, because $alpha_mle alpha_n le beta_n$. We can then consider $beta_m'+1$, and would have $beta_m'+1=beta_m'+((m'+1)^sttext dice)le beta_m'+nle alpha_m$, contradicting the maximality of $m'$. Therefore you have $0le alpha_m-beta_m'le n-1$ and the rest of the proof follows.






            share|cite|improve this answer












            The figure is mistaken, but the proof is not, after a small clarification. Ignore the figure.



            Winkler intended to assume $alpha_nle beta_n$. Furthermore, he intended $0$ to be an allowable index when choosing $m'$, where $beta_0=0$. This ensures $m'$ exists. For each $mge 1$, we have $alpha_m ge beta_0$. This implies that $i:0le ile n,alpha_mge beta_i$ is nonempty; possibly it only contains $i=0$. We then let $m'$ be the largest element of this set.



            By definition, $alpha_m-beta_m'ge 0$. If $alpha_m-beta_m'ge n$, then it must be that $m'<n$, because $alpha_mle alpha_n le beta_n$. We can then consider $beta_m'+1$, and would have $beta_m'+1=beta_m'+((m'+1)^sttext dice)le beta_m'+nle alpha_m$, contradicting the maximality of $m'$. Therefore you have $0le alpha_m-beta_m'le n-1$ and the rest of the proof follows.







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            answered Dec 11 at 18:41









            Mike Earnest

            20.1k11950




            20.1k11950











            • Thanks. I tried setting $1'=0,$ but somehow I couldn't make it work.
              – saulspatz
              Dec 11 at 18:49
















            • Thanks. I tried setting $1'=0,$ but somehow I couldn't make it work.
              – saulspatz
              Dec 11 at 18:49















            Thanks. I tried setting $1'=0,$ but somehow I couldn't make it work.
            – saulspatz
            Dec 11 at 18:49




            Thanks. I tried setting $1'=0,$ but somehow I couldn't make it work.
            – saulspatz
            Dec 11 at 18:49











            3















            It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong.




            Yes, take $alpha_nleq beta_n.$




            But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?




            Set $1'=0.$ When you take the intervening dice between two different "$m$"'s the lower "$m$" is excluded, so it's fine to use zero here. The higher "$m$" is included, but that's ok: if $alpha_m-beta_m'=alpha_M-beta_M'=c$ with $m<M$ then necessarily $M'>0$ because $beta_M'>beta_m'.$






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              3















              It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong.




              Yes, take $alpha_nleq beta_n.$




              But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?




              Set $1'=0.$ When you take the intervening dice between two different "$m$"'s the lower "$m$" is excluded, so it's fine to use zero here. The higher "$m$" is included, but that's ok: if $alpha_m-beta_m'=alpha_M-beta_M'=c$ with $m<M$ then necessarily $M'>0$ because $beta_M'>beta_m'.$






              share|cite|improve this answer























                3












                3








                3







                It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong.




                Yes, take $alpha_nleq beta_n.$




                But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?




                Set $1'=0.$ When you take the intervening dice between two different "$m$"'s the lower "$m$" is excluded, so it's fine to use zero here. The higher "$m$" is included, but that's ok: if $alpha_m-beta_m'=alpha_M-beta_M'=c$ with $m<M$ then necessarily $M'>0$ because $beta_M'>beta_m'.$






                share|cite|improve this answer













                It seems to me that there are two holes in this proof. The first is in the statement that the labels must be less than $n$. Suppose that $alpha_n-beta_nge n.$ Then $n'=n,$ and there is no larger index available. Then perhaps $alpha_nlebeta_n$ is right after all, and the diagram is wrong.




                Yes, take $alpha_nleq beta_n.$




                But this leaves the second problem, which doesn't depend on the relation between $alpha_n$ and $beta_n.$ Suppose that $a_1<b_1.$ How is $1'$ to be defined?




                Set $1'=0.$ When you take the intervening dice between two different "$m$"'s the lower "$m$" is excluded, so it's fine to use zero here. The higher "$m$" is included, but that's ok: if $alpha_m-beta_m'=alpha_M-beta_M'=c$ with $m<M$ then necessarily $M'>0$ because $beta_M'>beta_m'.$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 11 at 18:28









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