query: Proof of underlined numbers in grid
positive_passages.docid: 158094
positive_passages.text: Coloring 5 Largest Numbers in Each Row and Column Yields at Least 25 Double-Colored Numbers
negative_passages.docid: 112519
negative_passages.text: Line break in a modified proof environment (with show/hide feature)
negative_passages.docid: 27655
negative_passages.text: enumerated list in proof - margin, qed symbol
negative_passages.docid: 141101
negative_passages.text: Russia (2000) contest:Prove the existence of a pair of rows and columns with intersections differently coloured
negative_passages.docid: 7003
negative_passages.text: Inductive proof of a formula for Fibonacci numbers
negative_passages.docid: 25874
negative_passages.text: Showing that a certain map is not flat by explicit counterexample
negative_passages.docid: 2628
negative_passages.text: Counting number of moves on a grid
negative_passages.docid: 153544
negative_passages.text: Labeling backplanes in a 3D plot using "enclosed numbers" (i.e. ①, ② ...)
negative_passages.docid: 4303
negative_passages.text: Enumerating inside a proposition and inside a proof
negative_passages.docid: 39156
negative_passages.text: Concerning what is between two consecutive squares.
negative_passages.docid: 100565
negative_passages.text: Property of sup of a set of numbers
negative_passages.docid: 76029
negative_passages.text: Induction proof: n lines in a plane
negative_passages.docid: 34267
negative_passages.text: Grid System in LaTeX
negative_passages.docid: 128716
negative_passages.text: Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side
negative_passages.docid: 141482
negative_passages.text: Proof that plane with $N$ lines can be painted with two colors so that any two neighboring regions are painted in different colors
negative_passages.docid: 43033
negative_passages.text: Proof on Catalan numbers
negative_passages.docid: 83037
negative_passages.text: Applying Zorn's Lemma to natural numbers