# tiling Published articles for tiling. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## MLIR -- Running and Testing a Lowering DevFeed: [MLIR -- Running and Testing a Lowering]() Original publisher: [Read original article]() Published: 2023-08-10T14:36:13Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Compiler](), [Testing](), [Optimization](), [bazel](), [LLVM]() Tags: [bazel](), [c-plus-plus](), [compiler](), [compilers](), [fhe](), [llvm](), [mathematics](), [mlir](), [optimization](), [programming](), [testing](), [tiling]() ### AI overview This tutorial introduces MLIR dialects and lowerings, explains how MLIR incrementally transforms programs through intermediate representations, and demonstrates end-to-end testing for a simple lowering. ### Source excerpt Table of Contents Last time, we covered a Bazel build system setup for an MLIR project. This time we'll give an overview of a simple lowering and show how end-to-end tests work in MLIR. All of the code for this article is contained in this pull request on GitHub, and the commits are nicely organized and quite readable. Two of the central concepts in MLIR are dialects and lowerings. These are the scaffolding within which we can do the truly interesting parts of a compiler--that is, the optimizations and analyses. ## Tiling a Chessboard with Dominoes (Opposite Colors Removed) DevFeed: [Tiling a Chessboard with Dominoes (Opposite Colors Removed)]() Original publisher: [Read original article]() Published: 2011-11-18T10:32:54Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Algorithm]() Tags: [algorithm](), [analysis](), [chessboard](), [color](), [dominoes](), [proofs-without-words](), [tiling]() ### AI overview A constructive solution shows that a chessboard with two opposite-colored squares removed can be tiled with 2-by-1 dominoes. The article explains the placement pattern and notes that the corresponding algorithm runs in linear time. ### Source excerpt This is a natural follow-up to our first gallery entry on the impossibility of tiling certain chessboards with dominoes. Problem: Suppose we remove two squares from a chessboard which have opposite color. Is it possible to tile the remaining squares with 2-by-1 dominoes? Solution: Notice that if we remove two squares of opposite color, then there is only one way to place dominoes on the remaining squares according to this scheme (one cannot tile a domino across the "walls"). ## Tiling a Chessboard DevFeed: [Tiling a Chessboard]() Original publisher: [Read original article]() Published: 2011-06-26T20:24:09Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [color](), [standard]() Tags: [chessboard](), [color](), [dominoes](), [mathematics](), [standard](), [tiling]() ### AI overview The article uses coloring arguments to show that certain chessboard and grid regions cannot be tiled with dominoes. It first analyzes 2-by-1 dominoes on a chessboard with two opposite corners removed, then applies a specially chosen coloring to a 3-by-1 domino problem and generalizes the proof method. ### Source excerpt Problem: Take a chessboard and cut off two opposite corners. Is it possible to completely tile the remaining board with 2-by-1 dominoes? Solution: Notice that every domino covers exactly one white tile and one black tile. Counting up the colors, we have 32 white and 30 black. Hence, any tiling by 2-by-1 dominoes will leave two extra white squares unaccounted for. So no such tiling is possible. Problem: Cut one corner off a chessboard.