“All Ten Game” Explained: Gameplay, Scoring, and Pro Techniques

The all ten game is a daily math puzzle where you’re given four numbers and challenged to create expressions that equal every integer from 1 to 10. Using only those four numbers and basic operations, you try to “get all ten” targets. It’s simple to learn, surprisingly deep, and great for building problem-solving and arithmetic fluency.

Popularized by Beast Academy as a free daily math game, players use the four digits with operations like addition, subtraction, multiplication, division, and parentheses to build expressions for 1–10.

What Is the All Ten Game?

At its core, the all ten game is a math puzzle:

Goal: Use four given digits to create valid mathematical expressions that evaluate to each whole number from 1 through 10.

Key characteristics (based on the Beast Academy version):

• You’re given four digits (e.g., 3, 4, 8, 12).
• You may use +, −, ×, ÷, and parentheses.
• You must use all four digits in each expression, usually exactly once.
• Depending on the version, you may allow fractions and negatives.
• The challenge is to get all ten targets: 1, 2, 3, …, 10.

Some versions also allow concatenation (e.g., joining 3 and 4 to make 34) with certain restrictions.

Basic Rules of the All Ten Game

You can adjust the rules to fit your classroom, tutoring session, or game night, but this is a solid “standard” ruleset.

1. Setup

• Choose or receive four digits (e.g., 2, 5, 7, 9).
• Write the target numbers 1–10 in a column.
• Decide which rules you’re using:
  • Operations: +, −, ×, ÷
  • Extras (optional): fractions, negatives, concatenation

2. Creating Expressions

For each target (1 through 10), build a valid expression:

• Use all four digits each time.
• Use each given digit exactly once per expression.
• You may repeat operations, but not digits.
• Use parentheses to control order of operations.

Example (made-up digits):
Digits: 2, 3, 4, 6

• Target 1: (6÷3)−(4÷2)=2−2=0(6 ÷ 3) – (4 ÷ 2) = 2 – 2 = 0(6÷3)−(4÷2)=2−2=0 → not good for 1
• Target 1 (better): (6−4)−(3−2)=2−1=1(6 – 4) – (3 – 2) = 2 – 1 = 1(6−4)−(3−2)=2−1=1

3. Valid Operations (Standard Version)

Most all ten game variants allow:

• Addition (+)
• Subtraction (−)
• Multiplication (×)
• Division (÷)
• Parentheses for grouping

Optional extensions:

• Fractions (e.g., 4÷(2+2)4 ÷ (2 + 2)4÷(2+2))
• Negative values at intermediate steps
• Concatenation of original digits (e.g., 1 and 2 → 12)

4. Typical Variants

You’ll see slight rule differences depending on where you play:

• Beast Academy version: four given digits, basic operations, fractions and negatives allowed, new puzzle every day.
• AoPS / community solvers: often allow concatenation of original digits but not intermediate results.
• Classroom adaptations: teachers may restrict fractions or negatives for younger students.

Suggested Scoring System for the All Ten Game

The original all ten game is more of a puzzle than a “scored” game, but for families or classrooms, adding scoring makes it feel more like a competitive game.

Here’s an easy scoring system you can use.

Basic Scoring

• 1 point for each target number (1–10) you successfully build.
• Maximum 10 points per puzzle.

Timed Variant (For Advanced Players)

Add time for extra challenge:

• Base: 1 point per target.
• Bonus: +2 points if you get all ten in under 15 minutes.
• Penalty: −1 point if you use hints.

Example:

• You get 9 numbers → 9 pts.
• You take 12 minutes and no hints → 9 + 2 = 11 pts.

Classroom / Group Play

In a classroom setting you might:

• Run one daily puzzle as a warm-up.
• Award stickers or stars for anyone who gets 6+ targets.
• Give a “All Ten” badge for students who solve all 10 with clear, well-written expressions.

Step-by-Step: How to Play the All Ten Game

Here’s a simple flow you can use to teach the game.

Step 1: Write Down the Digits and Targets

• Example digits: 3, 4, 6, 8
• Targets: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Step 2: Start With “Easy” Numbers

Look for obvious things like:

• Sums (3 + 4 = 7)
• Differences (8 − 6 = 2)
• Simple products (3 × 4 = 12)

Then build around them using all four digits.

Step 3: Use Opposite Operations

If you get close but not exact, invert or undo an operation:

• Too big? Try division or subtraction.
• Too small? Try multiplication or addition.

Step 4: Don’t Forget Parentheses

Parentheses let you control which parts calculate first:

• (8−6)×(4−3)=2×1=2(8 − 6) × (4 − 3) = 2 × 1 = 2(8−6)×(4−3)=2×1=2
• 8÷(4−(6÷3))=8÷(4−2)=8÷2=48 ÷ (4 − (6 ÷ 3)) = 8 ÷ (4 − 2) = 8 ÷ 2 = 48÷(4−(6÷3))=8÷(4−2)=8÷2=4

Step 5: Move Strategically Through the Targets

Many players:

1. Start with 1, 2, 3 to get “wins” early.
2. Use those patterns to build larger numbers.
3. Tackle the “hard” ones (often 7 or 9) last.

Example Solutions for an All Ten Puzzle

Let’s walk through a small sample using digits 2, 3, 4, 6 with fractions and negatives allowed.

Note: There may be many solutions; these are just sample ideas.

You can see that building a full set 1–10 can get tricky fast — that’s exactly why the all ten game is such a good brain workout.

Pro Techniques for Mastering the All Ten Game

Once you know the basics, here are “pro-level” strategies to consistently get all ten.

1. Look for Reusable Patterns

Many puzzles have reusable building blocks:

• Make 1: aa\frac{a}{a}aa​ or bb\frac{b}{b}bb​
• Make 0: a−aa – aa−a or (aa)−1(\frac{a}{a}) – 1(aa​)−1
• Make 2: a÷a2a ÷ \frac{a}{2}a÷2a​

You can embed these patterns inside larger expressions.

2. Use Symmetry and Swapping

Operations are often symmetric:

• a+b=b+aa + b = b + aa+b=b+a
• a×b=b×aa × b = b × aa×b=b×a

If one combination doesn’t work, swap the roles of numbers or try reversing subtraction and division structures.

3. Think in Layers

Instead of trying to hit a target directly, think:

1. First make a useful intermediate (like 1, 2, or 5).
2. Then combine that intermediate with the remaining numbers.

Example:

• Digits: 2, 3, 4, 8
• Need: 10
  • Make 2: 4÷2=24 ÷ 2 = 24÷2=2
  • Use 8 and 3 to make 8: 8+3−18 + 3 – 18+3−1 (if 1 is available)
  • Or: 8+(4÷2)−3=8+2−3=78 + (4 ÷ 2) – 3 = 8 + 2 – 3 = 78+(4÷2)−3=8+2−3=7 → adjust further.

4. Use Fractions to Fine-Tune

Allowing fractions massively increases your options.

Examples:

• a+bc+d\frac{a + b}{c + d}c+da+b​
• abcd\frac{ab}{cd}cdab​
• a÷(b+c−d)a ÷ (b + c – d)a÷(b+c−d)

Fractions are especially helpful for awkward targets like 7 or 9.

5. Keep a Scratch List of “Mini-Results”

When solving, keep a quick list on the side:

• a+b=?a + b = ?a+b=?
• a−b=?a – b = ?a−b=?
• a×b=?a × b = ?a×b=?
• a÷b=?a ÷ b = ?a÷b=?

Having these on paper makes it easier to see connections.

6. Work Backwards From the Target

Ask yourself:

• “If I want 9, could I do 10−110 – 110−1, 3×33 × 33×3, or 182\frac{18}{2}218​ using these digits?”
• Then see if those sub-targets can be built from your digits.

This “reverse engineering” is a powerful pro technique in many arithmetic puzzles.

Educational Benefits of the All Ten Game

Educational platforms highlight games like all ten because they:

• Strengthen mental arithmetic (especially with four operations).
• Build number sense, fractions, and order-of-operations fluency.
• Develop persistence and creative problem-solving.
• Work well as low-floor, high-ceiling tasks — accessible to many levels.

Teachers often use it as:

• A warm-up activity
• A station in math centers
• An enrichment task for advanced students

FAQ About the All Ten Game

1. What exactly is the all ten game?

It’s a math puzzle where you use four digits and basic operations to build expressions that equal every integer from 1 to 10.

2. Do I have to use all four numbers every time?

In most versions associated with Beast Academy and AoPS, yes — you use each of the four digits exactly once in each expression.

3. Are fractions and negatives allowed?

Often, yes. Many write-ups explicitly allow fractions and negative intermediate results because they add richness and challenge to the puzzle.

4. Is there an “official” scoring system?

Not really — it’s usually a pure puzzle. The scoring systems in this article are optional variants you can use for class, clubs, or casual competition.

5. Where can I play a daily all ten game online?

Beast Academy hosts a daily All Ten puzzle as part of its advanced math curriculum for kids, and various teacher blogs and GitHub projects share printable versions and solvers.

Conclusion

The all ten game is a brilliant blend of simple rules and deep strategy: four digits, basic operations, and the challenge of building every number from 1 to 10. Whether you’re a teacher looking for a powerful warm-up activity, a parent wanting a brainy family puzzle, or a math enthusiast who enjoys number challenges, this game offers endless replay value.

By understanding the core gameplay, adopting a clear scoring system, and practicing the pro techniques above — like pattern spotting, using fractions cleverly, and working backwards from targets — you’ll steadily improve your ability to “get all ten” on even the toughest puzzles.