# Coded hundred square

## Problem

*Coded Hundred Square printable sheet*

This hundred square is written in code.

It starts with one and ends with a hundred.

Can you build it up? How did you do it?

Can you build it up in a different way?

Talk to a friend who has also tried building up the hundred square. How did they do it? What do you like about their method?

You can use the interactivity below or print and cut out the pieces from these sheets.

*You may be interested in the other problems in our Number Sense and Place Value Feature.*

## Getting Started

Where could you start?

What might the first row of numbers look like? Why?

How might the highest number look different from the others?

What do you know about the multiples of 11?

What will be the same in each column?

What will be the same for the first nine numbers in each row?

## Student Solutions

We had quite a number of submissions for this challenge that was a little different to many questions that get set.

Becky & Poppy from Little Thurrock Primary School sent in the following:-

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Ellie and Mirella from Westridge School United States sent in:

Solution for Coded hundred square: We thought #1 was the diamond because in the other pieces the first shape was the diamond. The second one was easy as pie, we calculated that the second one was the weird square thing with loops at the corners. The third threw us for a loop, we thought it was the empty circle. The fourth was the black circle and the fifth was the curly H. the sixth was the four iamonds making the shape of a bigger diamond. The seventh was the arrow with a line through it. The eighth was the two swirls. And the ninth was the empty square. Ten (Finally) was a big diamond and a smaller diamond next to it. And...Voila!! You've got your square!

Isabella Cuni from Wentworth Primary sent in her solution.

First of all you could start by looking for the singular codes. Then you could find which ones fit together. By this time you would of completed 1,2,3,4,5,6,7,8,9 and 10. The first numbers will stand out and will look like a singular figure. The last numbers might have the second to last figure in the tens column and then in the ones or units column the first nine figures. In the very last column, the figure might have in the hundreds column the first figure.

I know that the multiples of eleven - up to 99 - is always the same number e.g. 88, 44, 11, 66. In each column, the position of the ones are the sane e.g. 12,22,32,42 however the only thing that is different is they always have a different ten like 40 and 20 and the last number is different. The thing that will be the same in each nine numbers in each row is the ones are the same.

Brigitte, Daisy, Folayemi, Lyra, Olamide, Selah, Kenny, Moyo, Nathan, Ryan, Jordan and Juliana from Monarch Global Academy USA sent in some very good encouraging thoughts and ideas.

Brigitte

I saw that every number that had a 1 inside it is a diamond then I went on and on then I figured it out. Everything that had a two in it was that x shaped symbol. Everything that had a three in it was the white circle. I continued using that same pattern to find all the numbers. When I was done I checked my work to make sure that it made sense. It also fit inside the puzzle so I knew that I was correct.

Daisy

I saw all the way down there was diamonds so I knew 1 was diamond. I saw a small diamond where 0 should be so I knew 100 was big diamond small diamond small diamond. Then I started seeing rows and I started fitting pieces right as long as there was no gaps. it started looking perfect and I was done. It was all filled in so I must be done.

Folayemi

I solved by looking at the spaces and shapes to put the shapes out of the 100 square. When I was done it fit it just right, so I knew I was correct. I solved by having a strategy to look at the shapes and finding our which one represents 1 and which represents 10. I finally realized how to put them in the right place, and don't forget teamwork! I worked with Daisy and Brigitte. One more thing, I made mistakes but mistakes are part of learning. So I have to be careful!

Lyra

When you first start, you've got to try to choose a symbol for 1. Make sure that it will fit on the 100 chart without overlapping with any others. Then, keep going to try to figure out what 2 is. Keep on going on. If something doesn't make sense, then look at the symbols you already put on the chart and see if you got something wrong. Think about what other symbols could work. Once you get to the end and there's only a few more blank spaces then you can just think, "Ok, this fits here!" because you've already figured out what each of the symbols means.

Olamide

When you're doing it when you put your first block keep looking at the left side for the 11,21,31,41 and so on. Use the left number for the top row on the ones. for the next row use the left number every block that has the left number on the left use and put it next on the puzzle. Keep doing what I just said for the 20 row, 30 row,40 row and so on and you will get it.

Selah

My answer to Coded Hundred Square is to believe that you can do it, first of all. If you don't believe you can do it, you probably can't!! The second thing is logic. I think that if you really try, you can get it!!! Also, logic is key. Look at the square closely and analyze what number is the number one. Then look at the parts around it. Which other part looks like the part around yours? Keep working until you find the top row. Then try to figure out the left hand column that ends with 1. Once you have found those out, do the 10 column and the 90 row. Once you have figured out the top, bottom, left, and right parts, you should either be done or have to do the middle part. I chose this solution because when I used this, I got it right on the first try. I also want to say that when it gets hard, I believe in all of you, big or small, short or tall. Just remember, YOU CAN DO THIS! I feel like anyone can do this problem if they use a method similar to mine. I really tried hard and thought my method might work. Just believe in yourself. That's what I did, and I got it right on the first try!

Great words of encouragement!

Kenny

First, when you start, add a couple of pieces. Then you should try to figure out some numbers. Try some numbers like 1,2,3,4 and 5. Then, you could add more pieces to figure out the rest of the numbers. Look over to see if it makes sense, look for if the pieces fits well and the numbers makes sense. Remember this is a game of logic. Just believe in yourself! Look at your 100 chart to see if the numbers match. Remember, the number in the ones place will always be the same symbol as you look down the column. Look horizontally and vertically to see if you have the same symbols in each row and column. For example, 21, 31, 41... going vertically and 21, 22, 23... going horizontally.

Moyo

First, figure out what the 1 would be, so you can fill in the first column. Write down what you think 2 will be, so you can keep track. Then, figure out what all the other numbers would be. If you're not sure, try it out. See if the symbol works for the number you are trying out. I thought about the ones places and the tens place. When I figured what the big diamond represents, I knew where the ones would go on the other numbers like 11, 21, and 31. Also check to see if it will fit on the 100 chart. Then, once you think you're all done, recheck your work and make sure that it makes sense.

Nathan

Start by placing any one piece and think which one will fit. Before putting it there, think if any other shape could fit. Then, if not try another shape that could fit. After the one you choose is put on, and repeat until the coded hundred chart is filled. If you mess up, click on pieces to show the full piece and see if you can change a piece for another piece. Remember the first strategy to check if another one can fit after that one and then repeat again and again until you're done. After I fit the pieces together, I looked down each column. I checked that

all the symbols in the ones placed match. I looked across each row and checked that the symbols in the ones places were in order (1,2,3,4...).

Ryan

The way I found my answer was by trying to find the ones place symbol first. The ones place symbol will be the same in each row and column. You could either build a row at a time and see what pieces fall together, or build a column at a time. By finding the ones place symbols, I could then find the rest, like the 10s, 20s, 30s, and so on. You just have to make sure that all the ones make sense and the rest of the puzzle just falls to you.

Jordan and Juliana

One of the strategies we used to solve the Coded 100 Chart was to look for patterns. We did this by looking in each column and finding that the symbols end with the same shape. For example, the first column of numbers always end in a rhombus shape. The same thing with the rows. Each row, we noticed while putting the symbols together, begins with the same shape. To solve this problem, we think it's best to work with the shape you first put in the 100 Chart, and keep building off of that one. The first piece you out in the 100 Chart works best if it fits into a corner of the chart.

It was good to read the slightly different approaches that you had to solve this challenge. Thank you and well done.

## Teachers' Resources

**Why do this problem?**

This problem is an easily understood task so everyone can get started. It encourages children to think about the construction of the familiar hundred square and about the first one hundred numbers in our counting system. However, children may need to work on their resilience and perseverance skills as they
may discover it is not quite as straightforward as it first appears!

As they test out their ideas and become even more curious about the construction of this coded hundred square, encourage learners to offer conjectures, and to explain and justify their ideas.

Possible approach

*This problem featured in an NRICH Primary webinar in September 2021.*

You could introduce this problem first by asking the group to picture a 'standard' hundred square in their mind's eye. Challenge them to answer questions orally such as:

- Which number is in the top right-hand corner? [10]
- What is immediately below 10? [20]
- What is two squares to the left of 99? [97]
- I start on 34 and move three rows down and three places to the right. What do I land on? [67]

### Key questions

Where could we start?

What might the first row of numbers look like? Why?

How might the highest number look different from the others?

What do you know about the multiples of 11?

What will be the same in each column?

What will be the same for the first nine numbers in each row?

What could you do if you are stuck?

### Possible support

Some children may benefit from having a standard hundred square to refer to as they work on this problem. It might help to try this Hundred Square Jigsaw first.

### Possible extension

Learners could have a go at Hundred Square, which will further deepen their understanding of the structure of the number square and our number system generally. They might also like to try Alien Counting, which introduces different bases, or Which Scripts?, which looks at numbers in different languages.