Abstract: A non-transitory computer-readable medium storing instructions executable by a hardware processor to create a plurality of adaptive grids, present an adaptive hexagon grid in a drawing window on a display, prompt a user to create a first chemical structure on the adaptive hexagon grid, present a first plurality of atomic bond representations on the display, receive a first location selection chosen by the user from the adaptive hexagon grid, present a first plurality of atomic bond representations on the display, receive a first atomic bond selection chosen by the user from the first plurality of atomic bond representations, and present the first atomic bond selection at the first location selection on one of the plurality the adaptive grids. The first plurality of atomic bond representations include at least a single covalent bond representation and a double covalent bond representation.

Abstract: A back-end computing system configured to determine i) a first mathematical operation a user may carry out on a first mathematical expression and ii) a second mathematical operation the user may carry out on the first mathematical expression. To minimize processing loads on a user computing device, the system also provides to a mathematics application operating on the user computing device instructions to allow the user to attempt to carry out the first mathematical operation and the second mathematical operation. Further, the system determines i) an accurate outcome and an inaccurate outcome of performing the first mathematical operation and then provides the accurate outcome and the inaccurate outcome to the mathematics application. The system then receives an indication of a second mathematical expression from the mathematical application, where the second mathematical expression is based on the first mathematical operation and a selection of the inaccurate outcome by the user.

Abstract: A system includes a first chemical structure, a plurality of selectable start indicators, a plurality of selectable endpoint indicators, a first electron-pushing arrow, and an accuracy indicator. The first chemical structure includes at least one of a first chemical structure atomic bond indicator, a first chemical structure atom indicator, a first chemical structure lone pair indicator, and a first chemical structure functional group indicator. Each representation of the first chemical structure atomic bond indicator and each representation of an atom having the first chemical structure lone pair indicator includes a selectable start indicator. The first electron-pushing arrow includes a first tail associated with a location of one of the selectable start indicators and a first head associated with a location of one of the selectable endpoint indicators. The accuracy indicator indicates whether there was proper placement of electron-pushing arrow(s).

Abstract: A back-end computing system configured to determine i) a first mathematical operation a user may carry out on a first mathematical expression and ii) a second mathematical operation the user may carry out on the first mathematical expression. To minimize processing loads on a user computing device, the system also provides to a mathematics application operating on the user computing device instructions to allow the user to attempt to carry out the first mathematical operation and the second mathematical operation. Further, the system determines i) an accurate outcome and an inaccurate outcome of performing the first mathematical operation and then provides the accurate outcome and the inaccurate outcome to the mathematics application. The system then receives an indication of a second mathematical expression from the mathematical application, where the second mathematical expression is based on the first mathematical operation and a selection of the inaccurate outcome by the user.

Abstract: A non-transitory computer-readable medium storing instructions executable by a hardware processor to create a plurality of adaptive grids, present an adaptive hexagon grid in a drawing window on a display, prompt a user to create a first chemical structure on the adaptive hexagon grid, present a first plurality of atomic bond representations on the display, receive a first location selection chosen by the user from the adaptive hexagon grid, present a first plurality of atomic bond representations on the display, receive a first atomic bond selection chosen by the user from the first plurality of atomic bond representations, and present the first atomic bond selection at the first location selection on one of the plurality the adaptive grids. The first plurality of atomic bond representations include at least a single covalent bond representation and a double covalent bond representation.