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Problem 27 · The Knight’s Missing NumbersPrint one blank worksheet now, download its PDF, or prepare a family or classroom packet.

Problem 27

The Knight’s Missing Numbers

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The Knight’s Missing Numbers

Complete to collectIndicoliteLevel 3 · 5 Glints

The problem

On an empty 8 × 8 chessboard, a square’s knight-move count is the number of different squares a knight can reach from that square in one move.

A knight moves 2 squares in one direction, then 1 square in a perpendicular direction. Its landing square must stay on the board. The two center lines are mirror lines. Use the outlined 4 × 4 quarter to determine the counts for the whole board.

An empty 8 by 8 chessboard has two center mirror lines. The outlined upper-left 4 by 4 quarter has 16 blank circles, one per starting square. A reference diagram shows a knight in the center of a 5 by 5 grid with all eight possible L-shaped landing squares marked. The numbers 2 through 8 appear beside the board for the student to circle or cross out.
Map the outlined quarter, then decide which knight-move counts occur anywhere on the board.

Questions

Mark the map and number row on this page. Use separate paper for the short explanation.

  1. Write the knight-move count in each of the 16 blank circles in the outlined quarter. Use the knight-jump picture as a checklist so that you test every possible landing square exactly once.

  2. The numbers 2 through 8 appear beside the board. Circle every number that appears in your completed quarter map, and cross out every number that can never occur anywhere on the board. On separate paper, explain why reflecting the quarter across the two center lines cannot produce a new knight-move count.

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.

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 4

Uses Middle Grade 4 math.

Thinking

Stretch challenge

A nontrivial plan using familiar mathematical ideas.

Time & response

About 6–13 minutes

Moderate amount of written work · Investigation response.

How to support: The useful struggle is keeping track of all eight possible L-shaped jumps without missing or double-counting landings. Ask how the mirror lines help before discussing any missing numbers.

How did it feel?

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

Keep exploring

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Technical reference

Rating, skills & curriculum

Designed for Grade 4 · Uses Middle Grade 4 math · About 6–13 minutes

Grade 4 · Discrete mathematics and logic · Combinatorics

MJ 4.4 · C3 · W2

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 4.4

Grades 3–5

Grade 4 progression · point 4

Likely range
MJ 4.3–4.6
Confidence
80%

Challenge

C3 · Stretch

A nontrivial plan using familiar mathematical ideas.

  • Method selection1
  • Reasoning depth2
  • Novelty2
  • Constraint management2
  • Proof or justification2

Workload

W2 · Moderate

Several coordinated calculations, representations, or explanation steps.

Active time
6–13 minutes · typically 9
Response
Investigation
Known-method steps
18
  • Mechanical execution2
  • Written output1
  • Representation production2
  • Bookkeeping2

Learning focus

Skills this problem practices

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

  • Count outcomes systematically without omission or duplication

    The learner can count outcomes systematically without omission or duplication. This capability is distinct from mere exposure to or isolated use of tables, lists, multiplication principle.

    • Developing mastery
    • 100% focus
    • MJ 4.4 milestone

Readiness

Helpful prerequisites

Skills the rating assumes are already available to the learner.

  • 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.

    • Fluent readiness
    • Required
    • MJ K.5 milestone

External alignment

Curriculum and standards crosswalks

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

Common Core State Standards for Mathematics2010 · 1 location
Standard components1 location · Representative: Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.