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Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!


Age 11 to 14
Challenge Level

  This printable worksheet may be useful: Cryptarithms.

Why do this problem?

This problem builds on students’ fluency in adding integers and encourages them to choose possible integers based on their properties. Students will practice mathematical reasoning and systematic working.


Possible approach

Introduce questions in batches, starting with just questions 1-4 or even just 1 and 2. Ask students to work in pairs or groups to find answers, potentially stopping to give hints of which letter to focus on to begin each question.

Once students have attempted these questions, discuss their methods. Which number facts did they use? How did they know there weren't more answers? You might want to take notes on the board for students to refer back to later.

More questions can now be introduced in batches. Several of the questions - e.g 6 & 7, 11 & 12, 15 & 16, 17 & 18, 20 & 21 - pair together nicely and could be focused on together, if needed. There is opportunity to split these pairs amongst the class, which would allow for comparison between the differences in very similar questions.

Alternatively, all of the questions – perhaps without the extension – could be shown at once, with students allowed to choose which questions they attempt, either on their own or in pairs/groups.

Towards the end, the final challenge can then be offered. This can be attempted by all of the students, regardless of how far they have got with the other questions (although you might want to save it for later if the students have made little progress). You could pose the final challenge to the whole group, and allow students to share their best possible answer. Give students time to try to match and beat the best answers before asking those with the highest totals to share their methods.


Key questions

How can we prove we have found all of the solutions?

What number facts can we use?


Possible support

Replace one letter with its correct number in the questions. Begin by considering numerical facts, then introduce reasoning and systematic working.


Possible extension

Students could invent their own cryptarithms. How many solutions will their cryptarithm have? Can they create one with one, two, three, no solutions?

Two and Two continues the use of cryptarithms and extends ideas of mathematical reasoning.