Teaching 6174: a classroom lesson plan

Learning goals

• Practise ordering digits and four-digit subtraction, including borrowing across zeroes.
• Recognise an iterative process and describe it as a repeated rule.
• Understand the terms fixed point and edge case.
• Move from “it always seems to work” to “why does it work?”.

Starter (5 minutes)

Write 3524 on the board and demonstrate the routine once: 5432 − 2345 = 3087. Ask the class to continue. They should reach 8352 and then 6174. Ask one student to apply the routine to 6174 and let the class notice that it returns 6174.

Group investigation (15–20 minutes)

Give each group of three or four students a card with instructions:

1. Each person chooses a different four-digit number whose digits are not all the same.
2. Apply the routine until the result stops changing.
3. Record how many subtractions it took.
4. Compare results within the group and write down one thing you notice.

Collect iteration counts on the board as a tally chart. The class distribution will usually resemble the real one on the statistics page, with 3 as the most common count and 7 as the maximum.

Common difficulties

• Dropping leading zeroes. When a result is 999, it must be written as 0999. Otherwise students accidentally switch to the three-digit routine.
• Choosing repdigits such as 5555. These give 0000. Use this as a discussion point about why the rule needs “at least two different digits”.
• Subtraction errors. A single mistake can send a student into a long, confusing path. Pairs checking each other work well here.

Discussion questions (10 minutes)

• Why does changing the order of your starting digits not change the result after step one?
• Why is every result divisible by 9? (Hint: check the digit sums.)
• What happens with three digits? Predict first, then test.
• Is 6174 “found in nature”, or is it a property of how we write numbers?

The last question connects well with the article Is 6174 found in nature?.

Extension tasks

• Older students: show that the first difference equals 999(a − d) + 90(b − c). The full argument is in why every number reaches 6174.
• Find a number that needs exactly seven steps. (Examples: 0014, 1234 needs 3, but 2111 needs 5.) Then try the daily challenge.
• Try two digits and describe what happens instead of a fixed point.
• Compare with another iteration such as happy numbers or the 1089 trick.

Adapting the lesson

• Ages 10–12: focus on the routine itself, the tally chart and the idea of a number that “returns to itself”. Allow calculators for the subtraction after the first two rounds.
• Ages 12–14: add the three-digit routine, ask students to predict the destination before testing, and introduce the words iteration and fixed point.
• Ages 14–16: set the algebra 999(a − d) + 90(b − c) as the main task and ask students to explain why only 54 first results are possible.

Assessment ideas

• Exit ticket: “Explain in two sentences why 5555 does not reach 6174.”
• Error spotting: give a worked path containing one dropped leading zero and ask students to find and fix it.
• Short written task: “Describe the routine so precisely that someone who missed the lesson could follow it.” This checks both understanding and mathematical communication.

Answers for teachers

• Maximum number of steps for four digits: 7. For three digits: 6.
• Most common count: 3 steps (2,400 of 9,990 valid inputs).
• Every result is a multiple of 9 because a number and its rearrangement leave the same remainder when divided by 9.
• Two digits: the loop 09 → 81 → 63 → 27 → 45 → 09.