Math Puzzles Game: Logic Grids for Kids

What is a Calcudoku puzzle and how do you solve it?

A Calcudoku is a square grid where every row and every column holds each number from 1 up to the grid size once, and the bold-edged cages each carry a clue such as 7+ or 12× that the numbers inside the cage must reach. Start with the cages that have the fewest possibilities, such as a small sum or a product with few factor pairs, and list the number pairs that fit. Then use the row and column rule to cross pairs out. In Play you choose the puzzle type, tap a cell, tap a number, and use Pencil to jot down possible numbers. Check cell tells you whether one cell is right without giving the number, and Hint marks the row, cage or line to look at next.

📌 The Common Core practice standard on problem solving says that mathematically proficient students "analyze givens, constraints, relationships, and goals." (Common Core State Standards for Mathematics, Standard for Mathematical Practice 1)

Example: In a 4×4 grid, a two-cell cage marked 3+ can only hold 1 and 2, because 1 + 2 is the only way to make 3 from the numbers 1 to 4.

Try it: Free solve ▸

How do you solve a 4×4 or 6×6 sudoku?

A sudoku is a grid where every row, every column and every outlined box holds each number once, and a good puzzle has exactly one solution. In a 4×4 sudoku the boxes are 2×2 squares and the numbers are 1 to 4; in a 6×6 sudoku the boxes are 2 rows by 3 columns and the numbers are 1 to 6. Look for a row, column or box with only one empty cell, because the missing number is the one it does not have yet, then use each new number to find the next.

📌 McGuire, Tugemann and Civario list 288 completed 4×4 sudoku grids that need at least 4 clues, and 28,200,960 completed 6×6 grids that need at least 8 clues. (Gary McGuire, Bastian Tugemann and Gilles Civario, "There is no 16-Clue Sudoku", arXiv:1201.0749)

Example: A row of a 4×4 sudoku reads 3, 1, empty, 2, so the empty cell must be 4, the only number that row does not have.

Try it: Try sudoku ▸

Why does every row of a magic square add up to the same number?

A magic square is a grid of different numbers where every row, every column and both diagonals add up to the same total, called the magic constant. The whole square holds the numbers 1 to 9 once each, which add to 45, and the three rows share that total equally, so each row adds to 45 ÷ 3 = 15. A 4×4 square with the numbers 1 to 16 adds to 136, so each row adds to 136 ÷ 4 = 34. For a square of size n the constant is n × (n × n + 1) ÷ 2. Some squares in the game use numbers that count up by 2, 3 or more, and then the magic constant is shown above the grid.

📌 Weisstein notes that the only normal 3×3 magic square, which uses the numbers 1 to 9, was known to the ancient Chinese as the Lo Shu. (Eric W. Weisstein, CRC Concise Encyclopedia of Mathematics, "Magic Square")

Example: In the Lo Shu square, 2 + 7 + 6 = 15 along the top row, 9 + 5 + 1 = 15 along the middle row and 2 + 5 + 8 = 15 along a diagonal.

Try it: Try magic squares ▸

How do you solve a number pyramid?

A number pyramid is a stack of stones where each stone is the sum of the two stones directly under it, so the top stone holds the total pattern of the bottom row. You can work up by adding and down by subtracting. If a stone is 12 and the stone under it on the left is 5, the stone under it on the right is 12 − 5 = 7. Look for a small triangle of three stones where two numbers are known and fill in the third, then repeat with the new number.

📌 Grade 3 standard 3.OA.D.9 asks students to "Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations." (Common Core State Standards for Mathematics, 3.OA.D.9)

Example: A pyramid with the bottom row 3, 5, 4 has the middle row 8, 9 and the top stone 17, because 3 + 5 = 8, 5 + 4 = 9 and 8 + 9 = 17.

Try it: Try number pyramids ▸

How do you solve a puzzle one step at a time?

Look for the one cell, line or stone where only one number can fit, fill it in, and then use that new number to find the next one. Zayn's five stories (the cage code, shell sorting, the turtle's secret, the stone steps and the treasure map) each have three small puzzles with only a few empty cells, so you practice one clue at a time. Every story can be read aloud. Solve a step on the first check, with no hint and no wrong mark, and it counts toward three stars. A wrong check marks the cells to look at again but never shows the answer.

📌 The Common Core practice standard on structure says that mathematically proficient students "look closely to discern a pattern or structure." (Common Core State Standards for Mathematics, Standard for Mathematical Practice 7)

Example: In a 4×4 sudoku, a row reads 1, empty, 4, 3. The empty cell cannot be 1, 4 or 3, so it must be 2.

Try it: Zayn's stories ▸

Why does a good puzzle have exactly one solution?

A proper puzzle has exactly one solution, so logic alone always shows the next number and you never need to guess. The puzzle rounds give you one full puzzle of each type: Calcudoku, sudoku, a magic square and a number pyramid. "Just starting" uses 4×4 grids with + clues, 3×3 magic squares and 3-row pyramids; "I know the basics" uses 5×5 Calcudoku with + − ×, 6×6 sudoku and 4-row pyramids; "Challenge me" uses 6×6 Calcudoku with all four operations, fewer sudoku clues, 4×4 magic squares and 5-row pyramids. A computer solver tried every possibility for each puzzle in this game and found exactly one solution.

📌 McGuire, Tugemann and Civario proved by an exhaustive computer search that a standard 9×9 sudoku needs at least 17 clues to have exactly one solution. (Gary McGuire, Bastian Tugemann and Gilles Civario, "There is no 16-Clue Sudoku", arXiv:1201.0749)

Example: With only 7 clues a standard sudoku can never have one solution, because the two missing numbers could always be swapped.

Try it: Four rounds ▸

How do × and ÷ clues help you solve puzzles faster?

A × or ÷ clue works like a multiplication fact family: 12× in a two-cell cage means the cells multiply to 12, and 2÷ means the bigger number is twice the smaller one. Knowing your times tables turns a clue into a short list of number pairs at a glance. The 60-second sprint gives mini puzzles with 2 to 4 empty cells, keeps your streak, keeps your best score for each sprint length and brings back the puzzle types you miss more often. A wrong answer shows the right numbers marked ✓ with no penalty, and Skip moves on.

📌 Grade 3 standard 3.OA.C.7 asks students to "Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations." (Common Core State Standards for Mathematics, 3.OA.C.7)

Example: In a 6×6 grid a two-cell cage marked 2÷ can hold 1 and 2, 2 and 4, or 3 and 6, because in each pair one number is twice the other.

Try it: Mini puzzles ▸

Worked example: solving a 4×4 Calcudoku cage

1. Clue: two cells in one row form a cage marked 3+. The numbers are 1 to 4, and the other two cells of the row already hold 3 and 4.
2. The pair for 3+ is 1 and 2 (1 + 2 = 3), and the row rule agrees: only 1 and 2 are left for this row.
3. The left cage cell has a 2 above it in its column, so it cannot be 2. The left cell is 1 and the right cell is 2.
4. Check: 1 + 2 = 3, the row now reads 1, 2, 3, 4 and no number repeats.

Questions

Related games

• Times Tables Dojo — learn the facts behind × and ÷ cages
• Add & Subtract — practice the sums behind cages and pyramids
• Patterns & First Algebra — spot rules and patterns in numbers
• Factors & Primes — find the factor pairs for × clues

Published 12 Nov 2026 · Last updated 12 Nov 2026 · https://scisim.org/play/math-puzzles