There are $100$ points on the plane. All $4950$ pairwise distances between two points have been recorded. $(a)$ A single record has been erased. Is it always possible to restore it using the remaining records? $(b)$ Suppose no three points are on a line, and $k$ records were erased. What is the maximum value of $k$ such that restoration of all the erased records is always possible?
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Tags: geometry unsolved, geometry
AlphaBetaTheta
05.08.2011 07:12
anyone have any ideas for this one?
Mewto55555
06.08.2011 00:46
Are just the numbers recorded, or are which points they correspond to also recorded?
Goutham
06.08.2011 18:56
Mewto55555 wrote: Are just the numbers recorded, or are which points they correspond to also recorded? I guess, just the numbers are recorded.
Shu
06.08.2011 20:16
For part (a), the answer is no. E.g., let $S$ be the set of points \[ S=\{(0,1)\}\cup\{(0,0), (1,0), (2,0),\ldots,(98,0)\}. \] Then the distances assumed by the sets \[ A=S\cup\{(-1,0)\},\qquad B=S\cup\{(99,0)\} \] are identical except for one of them.
Shu
06.08.2011 21:07
I think a similar idea, but on a circle, can be used for part (b). Let $T$ be the set of vertices of a regular $102$-gon $P_0P_1\cdots P_{101}$. The distances assumed by the sets $T\setminus\{P_0,P_1\}$ and $T\setminus\{P_0,P_{2}\}$ are identical except for one of them.