# Gabriel's Problem

## Problem

*Gabriel's problem printable worksheet*

Gabriel wrote the numbers 1-9 in a 3x3 grid.

He then multiplied together all the numbers in each row and wrote the resulting product next to that row.

He also multiplied the numbers in each column together, and wrote the product under that column.

He then rubbed out the numbers 1-9.

**Can you work out where Gabriel placed the numbers 1-9?**

Did you have more information than you needed?

Can you fill these grids? One of them has more than one solution:

For this grid, Gabriel used the numbers 1, 2, 3, 4, 5, 6, 9, 10 and 12.

If you would like to try some more problems like these, then click on the button below.

Can you work out where to place the numbers

**1-12**in this grid?

Can you work out where to place the numbers

**1-12**in this grid?

Can you work out where to place the numbers

**1-12**in this grid?

Can you work out where to place the numbers

**1-16**in this grid?

Alternatively, you could try making some diagrams of your own, and give them to your friends to solve.**Wh****at features of a grid make it easier or harder to solve?****Extension:**

Gabriel used the numbers 1, 2, 3, 4, 6, 8, 9, 12 and 16 to make this grid.

How many solutions can you find?

Is this diagram easier or harder to complete than the ones above? Why?*Click here for a poster of this problem.If you enjoyed working on this problem you may also enjoy Product Sudoku.*

*With thanks to Gabriel Solari, whose idea formed the basis of this problem.*

## Getting Started

To get you started, the diagrams below record a potential route through each of the problems.

The numbers record the order in which the different cells could be filled in on the grids.

Of course, there are other possible routes through the problems.

The places where the number is in brackets indicates that there is a choice of number that can be placed there.

**1.**

**2.**

**3.**

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The number that you can work out is the product of the numbers in that column. The other column products might help you work this out.

**4.**

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**5.**

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**6.**

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**7.**

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**8.**There are a number of different routes through this problem, depending on what you do.

One possibility is to consider first where the number 9 might go, looking at all the rows and columns.

You might then want to consider where the number 16 might go. Thinking about the row DEF and the column CFI in particular might be useful here in narrowing down the options.

After that, try to see how much of the grid each combination lets you fill in!

## Student Solutions

We received lots of great solutions, with some excellent explanations, from students from Nower Hill High School.

Abishek sent this clear explanation of how he went about solving the first problem. Click on the image to see a larger version:

Here are Ben's approach, Simran's approach, Pranavan's approach, Holly's approach, Elisha's approach and Kania and Theivhikha's approach to the same problem. Rebecca also sent a clear explanation of her approach. You can see her work on the first two questions here.

Another student from Nower Hill High School said:

I tackled this problem by writing out the possible combinations of numbers whose products would equal the number on the side. After this, I would look at the ones with the least amount of combinations or the least common numbers so that I can fill them in. Also, having the combinations written at the side will allow me to see which ones cross over with each other so that
I could put them in the correct boxes for two calculations.

Avipsa tackled questions 1 and 3 by splitting the numbers at the end of the rows and columns into the products of their prime factors. Here are her solutions.

Alice from Suton Valence School started with numbers she knew had few factors. She said:

The product of the middle column is 21, which is only divisible by 1, 3 and 7. Therefore the numbers in the middle column had to be 1, 3 and 7.

The product of the left column and the middle row are multiples of 10 therefore they had to have 5 as one of their numbers. The only square they share is the middle square of the left column so that is where I put the 5.

Because 24 is not divisible by 7 I put 7 in the middle of the bottom row, leaving the 3 in the middle of the top row.

I then turned my attention to the middle row and calculated that 1 x 5 = 5 and 5 x 8 = 40, so I put the eight in the right-hand square of the middle row.

I then concluded that 9 was in the right-hand square of the bottom row because the products of the right-hand column and bottom row were high.

Well done to the students of St Louis School in Milan. We got several solutions from students there. Tommasso explained how he worked out where 5 and 7 went.

First of all I have looked at which numbers have numbers 5 or 7 as factors. Because 5 and 7 can be written just once, only two numbers have these numbers as factors and you can place numbers 5 and 7.

For example, in the first problem just 60 and 40 are multiples of 5 and, in the second problem, just 210 and 126 are multiples of 7.

Dividing the numbers on the same row and column by 5 and 7, you can work out the other numbers in the row and column. Once you found these numbers you can more easily find the places of the other numbers.

Two other students from Nower Hill High School had a go at one of the more difficult questions, that required them to place all the numbers from 1 to 12. Both Mia and Hera discovered that there was more than one possible solution to the problem.

Well done to Benjamin, Ian, Henry and Ryan from Donhead Preparatory School in the UK who worked on the most difficult puzzles. Here are some of their answers:

Davide explained how he used prime factors to work out where some of the numbers had to go. You can see his full solution here.

Gianluca thought about how he could create his own problem. He said:

I've found out that the products at the end of each row multiply to 362880 and the products at the end of each column multiply too to 362880.

Why must this be true? Can you use this fact to create your own problem?

## Teachers' Resources

The puzzles are available for classroom display as slides, in PowerPoint or PDF format.

This printable worksheet may be useful: Gabriel's Problem.

This spreadsheet may also be useful.

A poster of this problem is available here.