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Problem 14 · The Fire-and-Frost Dragon BridgePrint one blank worksheet now, download its PDF, or prepare a family or classroom packet.

Problem 14

The Fire-and-Frost Dragon Bridge

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The Fire-and-Frost Dragon Bridge

Complete to collectPerfect Round PearlLevel 5 · 8 Glints

Two fire dragons start on the two left stones and face right. Two frost dragons start on the two right stones and face left. The middle stone is empty. The goal is to swap sides, with the middle stone empty again.

Move one dragon at a time. A dragon may slide one stone forward into the empty space. Or it may jump forward over exactly one dragon of the other kind into the empty space. A dragon may never move backward or turn around.

Start: two fire dragons face right on the left two of five stones, the middle stone is empty, and two frost dragons face left on the right two stones. Goal: the frost dragons have the two left stones, the middle is empty, and the fire dragons have the two right stones.

Slide

Move forward onto the next empty stone.

Jump

Leap over one other-kind dragon into the empty stone.

Forward only

No backing up.
No turning around.

Dragon challenge

Use the investigation record. Continue the proof on separate paper as needed.

  1. What is the fewest number of moves that can get all four dragons to the goal?

  2. Find every different shortest path from the start to the goal. Show where the empty stone is after every move. Two paths are different if the empty stone is in a different place after any move.

  3. Explain why your list has every shortest path and why there cannot be another. You may use pictures, numbers, or words.

Investigation record

State log
MoveFive-stone stateEmpty stone
Start
1
2
3
4
5
6
7
8
9
10
Shortest paths and branch check
PathEmpty-stone sequence
A
B
C
D

Completeness notes: At each branch, list every legal forward move and explain why any rejected move cannot belong to a shortest solution.

At a glance

About this problem

These tags show what the problem is about and the math you’ll use. Choose one to find similar problems.

Theme

Skills

Helpful first

Grade

Math area

Topics

Is this a good fit?

Before you choose this problem

A plain-language look at readiness, thinking, time, and the expected response.

Math readiness

Designed for Grade 5

Uses Late Kindergarten math.

Thinking

Olympiad challenge

A sustained original strategy with several significant insights and proof.

Time & response

About 12–30 minutes

Heavy amount of written work · Proof response.

How to support: This is an older-student exhaustive investigation; encourage a state table and require a completeness argument.

How did it feel?

Pick the option that best matches this challenge. Recommendations use topic, math area, grade, story, and thinking challenge.

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Technical reference

Rating, skills & curriculum

Designed for Grade 5 · Uses Late Kindergarten math · About 12–30 minutes

Grade 5 · Number and quantity · Operations and computation · Geometry and spatial reasoning · Counting, cardinality, and numerals · Place value and numeration · Operation foundations · Shapes and spatial reasoning

MJ K.7 · C5 · W3

Precise MJD coordinates, factor scores, skill weights, calibration evidence, and curriculum crosswalks. This reference is available to everyone and is intentionally last.

Content level

MJ K.7

Grades K–2

Kindergarten progression · point 7

Likely range
MJ K.5–K.9
Confidence
84%

Challenge

C5 · Olympiad

A sustained original strategy with several significant insights and proof.

  • Method selection3
  • Reasoning depth3
  • Novelty3
  • Constraint management3
  • Proof or justification3

Workload

W3 · Heavy

Sustained mathematical work and a substantial written response.

Active time
12–30 minutes · typically 18
Response
Proof
Known-method steps
18
  • Mechanical execution2
  • Written output3
  • Representation production3
  • Bookkeeping3

Learning focus

Skills this problem practices

What a successful response demonstrates, with each skill’s share of the content rating.

  • Count objects with one number word per object

    The learner can count objects with one number word per object. This capability is distinct from mere exposure to or isolated use of stable order principle.

    • Independent mastery
    • 10% focus
    • MJ K.2 milestone

  • Use the last count word as the total

    The learner can use the last count word as the total. This capability is distinct from mere exposure to or isolated use of one-to-one counting.

    • Independent mastery
    • 10% focus
    • MJ K.3 milestone

  • Count up to 20 arranged objects and up to 10 scattered objects

    The learner can count up to 20 arranged objects and up to 10 scattered objects. This capability is distinct from mere exposure to or isolated use of one-to-one counting.

    • Independent mastery
    • 10% focus
    • MJ K.5 milestone

  • Compare groups as greater, less, or equal

    The learner can compare groups as greater, less, or equal. This capability is distinct from mere exposure to or isolated use of count or match objects.

    • Independent mastery
    • 10% focus
    • MJ K.5 milestone

  • Compose and decompose within 5

    The learner can compose and decompose within 5. The capability is credited only when the task makes that demand necessary for success.

    • Independent mastery
    • 35% focus
    • MJ K.7 milestone

  • Use relative-position words to describe shapes and objects

    The learner can use relative-position words to describe shapes and objects. This capability is distinct from mere exposure to or isolated use of spatial vocabulary.

    • Independent mastery
    • 25% focus
    • MJ K.3 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • Count objects with one number word per object

    The learner can count objects with one number word per object. This capability is distinct from mere exposure to or isolated use of stable order principle.

    • Independent readiness
    • Required
    • MJ K.2 milestone

  • Use the last count word as the total

    The learner can use the last count word as the total. This capability is distinct from mere exposure to or isolated use of one-to-one counting.

    • Independent readiness
    • Required
    • MJ K.3 milestone

  • Count up to 20 arranged objects and up to 10 scattered objects

    The learner can count up to 20 arranged objects and up to 10 scattered objects. This capability is distinct from mere exposure to or isolated use of one-to-one counting.

    • Independent readiness
    • Required
    • MJ K.5 milestone

  • Compare groups as greater, less, or equal

    The learner can compare groups as greater, less, or equal. This capability is distinct from mere exposure to or isolated use of count or match objects.

    • Independent readiness
    • Required
    • MJ K.5 milestone

  • Compose and decompose within 5

    The learner can compose and decompose within 5. The capability is credited only when the task makes that demand necessary for success.

    • Independent readiness
    • Required
    • MJ K.7 milestone

  • Use relative-position words to describe shapes and objects

    The learner can use relative-position words to describe shapes and objects. This capability is distinct from mere exposure to or isolated use of spatial vocabulary.

    • Independent readiness
    • Required
    • MJ K.3 milestone

External alignment

Curriculum and standards crosswalks

Approved evidence-backed locations that support discovery. They do not define the MathJewels rating.

Bridges in Mathematicsthird-edition · 50 locations
Modules24 locations · Representative: Module 1: Sorting Shoes
Number Corner workouts26 locations · Representative: September: Calendar Collector
Common Core State Standards for Mathematics2010 · 6 locations
Standards4 locations · Representative: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
Standard components2 locations · Representative: When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.