pigeonhole principle q (1 Viewer)

[eternallyboreduser]

Would my working out still be accepted? Because the written solution is entirely different but I thought that the lattice points would be evenly spaced out, no?

 

[eternallyboreduser]

 

[liamkk112]

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i think the problem with ur answer is that you've assumed that the points are of the form (a, b) and (5a, 5b). but what about if i picked the points (1, 2) and (3,4)? both are lattice points, but are not of the form (a, b) and (5a, 5b)

 

[eternallyboreduser]

i think the problem with ur answer is that you've assumed that the points are of the form (a, b) and (5a, 5b). but what about if i picked the points (1, 2) and (3,4)? both are lattice points, but are not of the form (a, b) and (5a, 5b)

Oh okay fair enough i understand their working now lol

 

[eternallyboreduser]

Oh okay fair enough i understand their working now lol

Holdup @liamkk112 I'm kinda confused on the last bit, where it says "but we have 5 points..." and onwards, do you mind explaining? thanks

 

[liamkk112]

Holdup @liamkk112 I'm kinda confused on the last bit, where it says "but we have 5 points..." and onwards, do you mind explaining? thanks

let (a,b) represent any lattice point. then as a,b are integers, there are four forms for (a,b): either (odd, odd), (even, even), (odd, even) or (even, odd). as we have five points of the form (a,b), by the pigeonhole principle at least two points will be of the same form. so let (c,d) represent the other point of the same form as (a,b). then for the midpoint of the line segment between these two points to be an integer, we need a+c and d+b to be an even integer since when we divide them by 2 we need an integer also, as lattice points are integer valued points. even + even is even, odd + odd is even. (a,b) and (c,d) are of the same form so a, c are of the same parity and b,d are of the same parity. hence a+c and b+d are both even, so the midpoint of the line segment between the two points will be a lattice point as both (a+c)/2 and (b+d)/2 are integers

 

[eternallyboreduser]

let (a,b) represent any lattice point. then as a,b are integers, there are four forms for (a,b): either (odd, odd), (even, even), (odd, even) or (even, odd). as we have five points of the form (a,b), by the pigeonhole principle at least two points will be of the same form. so let (c,d) represent the other point of the same form as (a,b). then for the midpoint of the line segment between these two points to be an integer, we need a+c and d+b to be an even integer since when we divide them by 2 we need an integer also, as lattice points are integer valued points. even + even is even, odd + odd is even. (a,b) and (c,d) are of the same form so a, c are of the same parity and b,d are of the same parity. hence a+c and b+d are both even, so the midpoint of the line segment between the two points will be a lattice point as both (a+c)/2 and (b+d)/2 are integers

yeah thats what i thoughg but it says odd even plus odd even, why is it narrowed down to that one specifically?

 

[liamkk112]

yeah thats what i thoughg but it says odd even plus odd even, why is it narrowed down to that one specifically?

i think they put dots there to signify the same thing happens in all cases, but u are right that it doesn’t have to be that case. but in all cases it works out

 

[eternallyboreduser]

i think they put dots there to signify the same thing happens in all cases, but u are right that it doesn’t have to be that case. but in all cases it works out

Oh okay yeah i understood it then, it was just the last bit that confusrd me a bit thanks!