Why 6174 Is Kaprekar's Constant

Introduce the descending-minus-ascending procedure and show a single worked example so the learner sees what 'repeating the routine' means.

• Take a 4-digit number with at least two different digits.
• Form the largest and smallest numbers by sorting its digits.
• Subtract the smaller from the larger and repeat.

Let the learner enter a 4-digit number and watch the routine iterate, so they commit to a guess before the structural argument appears.

• Type any 4-digit number with at least two distinct digits.
• Predict whether it ends at 6174, at 0, or at a different value.
• Run the iteration and observe the actual landing point.

Present descent traces from several different starting numbers to show empirically that they all funnel into 6174.

• Try 2111, 9830, 1000, 4994 — all are routed to 6174.
• Maximum observed descent length is 7 steps.
• Numbers with all digits identical become 0 and are excluded from the map.

A small directed graph widget that visualizes reachable values and their transitions, so the learner can see 6174 as the unique sink.

• Each node is a 4-digit value with at least two distinct digits.
• Directed edges show the result of one Kaprekar step.
• 6174 is shown with a self-loop; no node leads to a different fixed point.

Lay out the core proof idea: finite set, deterministic step, so every orbit must terminate in a cycle — and the cycle must contain 6174.

• At most 9000 reachable 4-digit values form a finite set.
• A deterministic map on a finite set makes every orbit eventually periodic.
• 6174 is the only periodic element reachable from valid starts.

A highlighter widget where the learner selects a starting node and the simulation traces its orbit forward step by step, illuminating each edge until it reaches 6174.

• Click any node to highlight its orbit.
• Watch the path collapse through small chains before reaching 6174.
• Confirm that 6174 has only a self-loop as outgoing edge.

A single transfer question asking the learner to predict the analogous constant for 3-digit numbers based on the 4-digit mechanism just explained.

• Apply the same descent argument to 3-digit numbers.
• Identify the 3-digit Kaprekar constant by reasoning, not recall.

Show that the same structure fails or changes for inputs the procedure was not designed for: uniform-digit numbers, 3-digit starts, and 5-digit starts.

• Numbers like 1111 map to 0, not 6174.
• For 3-digit starts the attracting constant is 495.
• For 5-digit starts there is no single universal attractor.

Tie the prediction, evidence, and finite-set argument together to directly answer the driving question.

• 6174 equals 7641 minus 1467, so it is a fixed point of the map.
• It is the only fixed point reachable from valid 4-digit starts.
• Every valid start descends to 6174 in at most 7 steps.