Crosses
Years 4 - 8

Crosses being explored

in a Year 6 class in Eritrea

guided by Australian Aaron Peeters.

Preparation

  • Tear a piece of paper into nine (9) pieces about the same size.
  • Write the digits 1 to 9 on the pieces, one on each piece of paper
  • Write the title of this challenge and today's date on a fresh page in your maths journal.
If the whole family want to join in this activity then use nine A4 pieces of paper and write big numbers like the ones on the board in the photo.

Getting Started

  • The first challenge is to put the numbers 1 to 9 in the spaces so that the horizontal line and the vertical line add to the same number.
  • Keep trying until you can do it, then draw the solution in your journal.

The Next Challenge

  • Find two (2) more ways to do it.
  • Write the solutions in your journal.
Have fun exploring Crosses.

The Big Challenge

Now you have found three solutions. Are they really different? The answer to that depends on how you answer these questions.

  • Do you think a horizontal or vertical line of 3 + 7 + 2 + 6 + 4 is different from 7 + 3 + 2 + 6 + 4?
  • Do you think a horizontal or vertical line of 3 + 7 + 2 + 6 + 4 is different from 4 + 3 + 2 + 6 + 7?
  • Do you think a horizontal or vertical line of 3 + 7 + 2 + 6 + 4 is different from 7 + 3 + 4 + 6 + 2?
Write your answers in your journal and give reasons.
Explain your rules for deciding when a solution is different.
  • Find one more different solution.
There might be more. At this point a mathematician would ask:
  • How many different solutions are there?
  • How will I know when I have found them all?
Look at the strategies on your working like a mathematician page.
  • Which strategy do you think your mathematician might choose to try to find more solutions?
Using the strictest definition of solutions being different, there are eighteen (18) different solutions...
...and 1,152 variations for each of them.
Find as many different solutions as you can and record them.
  • This can be a project for the family to tackle over time. Perhaps you can work with another family and keep a record on one of the social media apps.
There are hints in the Answers & Discussions that might help.

Digging Deeper

  • What happens if the cross arms are only three (3) squares and we use the digits 1 to 5?
    How many solutions are there? (Perhaps none.)
    How will we know when we have found them all. (If there are any.)
  • What happens if the cross arms are seven (7) squares and we use the numbers 1 to 13?
  • If there are solutions for 5, 9, 13, ... arm crosses, is there a pattern in the minimum number of solutions as the cross gets bigger?
  • Suppose we use the original cross and the digits 0 to 8. Are there any solutions?
  • Let's do an experiment.
    • Suppose all nine digits cards were placed face down and mixed around before they were put into a cross shape still face down.
    • The experiment is that we turn them over to see if they are a solution to the Crosses puzzle.
    • Suppose the experiment is done 30 times.
    • What chance do you think there would be of a solution? 0 chances in 30, 1 chance in 30, 2 chances in 30, ...?
    • Predict then carry out the experiment.

Note: If your school is a member of Maths300, your teacher can provide software that will help you explore Crosses.

Just Before You Finish

Read your Working Like A Mathematician page again and answer this question in your journal.
  • What do you know now about Working Like A Mathematician that you didn't know before you started Crosses.

 

Answers & Discussion

Hints
  • Is there anything special about the middle number?
  • Can all the nine digits be used in the middle?
  • Suppose you put a digit that works into the middle. What has to be done with the other eight (8) digits?
  • Does it help to know that the sum of the digits 1 to 9 is 45?

These photos from Year 4 at Homer Pittard School, Tennessee, show some of the solutions.

These notes were originally written for teachers. We have included them to support parents to help their child learn from Crosses. Perhaps the best thing about the notes for Crosses link is that it includes a story from Ulla Öberg in Sweden which proves that this number puzzle was the real work of a real, professional mathematician.

Send any comments or photos about this activity and we can start a gallery here.

 

Maths At Home is a division of Mathematics Centre