apps for education

Kids Num

Learn first numbers with kids! Count, listen, read.

Three the most popular languages in the world by native speakers (wikipedia): Chinese, Spanish, English.

Arabic and Roman numerals.

Arranged in Latin square 4x4 (as a little Sudoku). Latin square in wikipedia.

Periodical testing is also fun: turn 🤔 into 🤩.

Works perfect at any device orientation, portrait or landscape - numbers always upright.

Sound:

Ring/Silent switch on iPhone controls speaker (headset is always ON, adjust volume)
Background melodic ostinato momentarily can be turned ON/OFF via the 'settings' button ("gear" icon).

CT Lite / CT Lite+

Introduce kids to integrals

The puzzle informally and playfully introduces the concept of an integral, giving kids a first impression of it as the area under a curve. "The discrete equivalent of integration is summation" (wikipedia)

Example:

Play involves practicing summation: 1 + 1 + 1 = 3, 1 + 1 + 1 + 1 = 4, 1 + 1 = 2 (count orange squares in each column, the rightmost just 1). Finally: 3 + 4 + 2 + 1 = 10 (total number of orange squares)

CT Lite (formerly CT Kids) is easy-mode puzzle. CT Lite+ is upgraded version.

... from 1D to 2D

Considering example above as a binary function on compact support, the occupied area can be calculated either way: 3 + 4 + 2 + 1 = 4 + 3 + 2 + 1 = 10. Each clue in turn can be considered as a marginal integral over x or y (discrete equivalent).

Why Integrals Matter:

1) Integration is a fundamental operation in calculus, essential for understanding advanced mathematical concepts.

2) The rectangle method involves breaking the region under a curve into a series of rectangles to approximate the area.

... from 2D to 3D (Figure with red cell)

Let's assume red block denotes value 2 (in contrast to orange blocks, each of them is 1). Considering this example as a step-wise function, the total volume can be calculated either way: 3 + (3 * 1 + 1 * 2) + 2 + 1 = 4 + (2 * 1 + 1 * 2) + 2 + 1 = 11. This is again a discrete equivalent of integration.

CT Lite+ introduces gradients:

Each column and row has its own value ("weight"). Clues are now the weighted sums along the vertical and horizontal directions.

... from integral to mass distribution

Now each clue represents a first moment over x or y : integrals of x*f(x,y) or y*f(x,y), its discrete equivalents.

In physics, the first moment is closely related to the center of mass — balance point of a mass distribution.

xc = (7 + 4 + 10 + 6) / 9 = 2.7 (counting from top)

yc = (3 + 7 + 4 + 10) / 9 = 3 (exactly middle of the third column)

How to play Gradients puzzle (part of CT Lite+ )

Rules: Each cell has a value that contributes to the sum when calculated along a row or a column. The values (1, 2, 3, 4, 5) are printed above and to the left of the grid. The shaded cells participate in the summation, while the unshaded cells do not. The sums are weighted: each shaded cell contributes its own value ("weight"). Additionally, the value of each cell depends on the direction of the summation (either along the row or along the column). The dependence of the summation direction creates an anisotropic field combining horizontal and vertical gradients.

For example, the top-left corner is worth 1, and the bottom-left cell is worth 5 when summing along the left column. However, the same bottom-left cell is worth 1 when summing along the bottom row.

The grid starts empty. Below and to the right of the grid, the clues are printed. These clues represent weighted sums: the sums along the rows are printed to the right of the grid, and the sums along the columns are printed below the grid. To solve the puzzle, the player must shade some cells in a way that satisfies all the clues. (Note: the trivial case where all sums are zero is not considered interesting.)

... little bit more in simple words

When we calculate the sum of the shaded cells, the value of each cell depends on the direction we're summing — either horizontally (across rows) or vertically (down columns). This means that the same cell can have different values depending on the direction of the sum. This creates what's called an anisotropic field, where the value of a cell changes in different directions. The word 'anisotropic' just means that something behaves differently in different directions. In this case, the value of the cell is different depending on whether you're summing along a row or a column. This gradual difference in values creates two gradients: a horizontal gradient (across the rows) and a vertical gradient (down the columns), where the values gradually increase or decrease as you move along the rows or columns.

Examples how to solve Gradients

Easy case

Identifying certain fills: shade the cell with value 1 in the row with clue 1, same approach for cell 2 in the middle column. Mark ("-") the other cells in solved row and column ("mark" or "flag" means our logic dictates that no shade will be there). Next, set 2+4 in the row labelled 6 (and incidentally the column with clue 3 happens to match). Then, shade bottom-right corner: it inevitably fit 5 in horizontal sum and 5 in vertical sum. Finally, set 4 in topmost row. Puzzle solved.

Hard case

There are no rows and no columns with clues 1, 2, 15 (all shaded), 14 (15-1), 13 (15-2). Allora, we cannot shade for sure any cell so far... But we can mark (means "no shade can be here") two bottom cells (4 and 5) in rightmost column — since clue 3 can be satisfied only as 1+2 or 3. After that we can fulfill the bottom row — since 9 without 5 can be decomposed only as 2+3+4. Now we can shade cell 2 in the leftmost column since two possible decompositions of 6 consist it: 1+3+2 or 4+2. Next move: to shade 4 in the row labeled 10, because two only possible decompositions have it: 1+5+4 or 1+2+3+4.

The rest is easy. Fulfill the column with clue 10 (one variant only). Now we can complete simultaneously and independently two rows labelled 8 and 5 (and columns with clues 9 and 12 happen to match automatically). Finally, finish simultaneously columns with clues 6 and 3. Hard puzzle solved.

***

Even more hard cases can appear. Sometimes, the clues can be satisfied by different patterns of shaded cells. This means that, without additional information ( CT Lite+ provides essential clue in Competition Mode ), solving the puzzle may involve a guessing step. A player may suddenly face the need to decide how to shade a group of cells while considering how that choice affects another subset of cells. In a logically unambiguous puzzle, a wrong guess leads to a 'dead end' within a few steps, where the player encounters a contradiction between the clues and must backtrack. However, in randomly generated puzzles, different guesses can yield multiple non-contradictory solutions, each satisfying the clues.

An interesting phenomenon is the formation of 'isomers,' where an alternative solution can be created by inverting a subset of cells (changing all shaded cells to unshaded and vice versa). When playing for fun, any of these isomers is considered a valid solution.

Here is an example of ambiguous clues. Isomers appear due to group of cells 2+3 = 5 in rows and 1+2=3 in columns.

CT Lite+ (formerly CT Kids+) app

CT Kids. Versions 1.0 - 1.1 (May 2024 - January 2025). 4x4 board, two projections, no rotation. Easy to start: Welcome to CT world!

CT Kids+ was rebranded to CT Lite+ (available since February 2025). Version 2.0 combines 4x4 (= CT Kids) and 5x5 "gradients" boards. Still no rotation, two and two+ (in CT Lite+) projections. Two+ means: horizontal sums, vertical sums, and 1 diagonal sum — appeared sometimes on 5x5 board.

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