Royal Institution Primary Maths Masterclasses Off the shelf

Royal Institution Primary Maths Masterclasses Off the shelf Masterclass: Magic Squares rigb. org @Ri_Science Image credits: Ad Meskens via Wikimedia Commons, Melancholia I by Durer

What is a Magic Square? • A square grid with a number in each box • All the columns add to give the same number • All the rows add to give the same number • All the diagonals add to give the same number • This number is called the “Magic total” • Some squares are even more magic than that, but more of that later… • Let’s start by making a 3 x 3 magic square using the numbers 1, 2…, 8, 9

Working in pairs: • Use the digit cards to make a 3 x 3 square where each of the rows, columns and diagonals add to a total of 15 • When you have found a solution, draw it on the board • Now try to find another, different solution. (Think about how you will decide whether your new solution is different. ) • How many different solutions can you find?

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The CHRISTMAS LECTURES are the Ri’s most famous activity and are televised on the BBC. The first maths lectures by Prof. Sir Christopher Zeeman in 1978 started off the Masterclass programme! Christmas Lecturers include Michael Faraday, David Attenborough, Carl Sagan, Richard Dawkins, Alison Woollard, Saiful Islam & Alice Roberts Image credits: Tim Mitchell, Paul Wilkinson

Royal Institution videos • CHRISTMAS LECTURES – on the Ri website

Royal Institution videos • CHRISTMAS LECTURES – on the Ri website • Ri on You. Tube – experiments, videos & talks for all ages

Royal Institution videos • CHRISTMAS LECTURES – on the Ri website • Ri on You. Tube – experiments, videos & talks for all ages • Expe. Rimental – science experiments at home

Royal Institution Primary Maths Masterclasses Off the shelf Masterclass: Magic Squares rigb. org @Ri_Science Image credits: Ad Meskens via Wikimedia Commons, Melancholia I by Durer

What is a Magic Square? • A square grid with a number in each box • All the columns add to give the same number • All the rows add to give the same number • All the diagonals add to give the same number • This number is called the “Magic total” • Some squares are even more magic than that, but more of that later… • Let’s start by making a 3 x 3 magic square using the numbers 1, 2…, 8, 9

Working in pairs: • Use the digit cards to make a 3 x 3 square where each of the rows, columns and diagonals add to a total of 15 • When you have found a solution, draw it on the board • Now try to find another, different solution. (Think about how you will decide whether your new solution is different. ) • How many different solutions can you find?

Can you see the link between the left hand squares and the right hand squares? Look carefully at each pair. Try to find a single rule which links the left to the right for each row. 8 1 6 2 9 4 3 5 7 7 5 3 4 9 2 6 1 8 8 3 4 2 7 6 1 5 9 9 5 1 6 7 2 4 3 8 6 7 2 9 5 1 1 5 9 2 7 6 8 3 4 4 9 2 6 1 8 3 5 7 7 5 3 8 1 6 2 9 4

Other ideas to explore: • Can you make a new magic square if you swap the 1 for a 10? (So you are using the numbers 2, 3, 4, 5, 6, 7, 8, 9, 10) What is the magic total now? 18 • What consecutive numbers would you need to make a magic square with the magic total of 21? (3 -11) • What numbers could you use to make a magic square with a magic total of 150? (Yes- make all numbers 10 x bigger… 10, 20, 30…. 90) • Could you make a magic square with a magic total of 45? (Yes- make all numbers 3 x bigger… 3, 6, 9…. 30) • Could you make a magic square with a magic total of 16? (Not if sticking to 3 x 3 square with consecutive numbers: this will always have multiple of 3 as magic number) Explain your answer.

Do you recognise these images? What country do you think they might be from? They are over 3000 years old… Image credits. By Anon. Moos - Own work - Made by self from scratch, following layout of PD image File: Luo 4 shu 1. jpg. , Public Domain, https: //commons. wikimedia. org/w/index. php? curid=8888999

“Melancolia” by Albrecht Durer • German artist lived 14711528 • 20 years younger than Leonardo Da Vinci • What mathematical images can you see in the picture? • Durer was interested in the links between art and maths Image credits: Ad Meskens via Wikimedia Commons

Image credits: Ad Meskens via Wikimedia Commons

What do you notice about the numbers in the magic square? What is the magic total? 34 Find some pairs which sum to 17. How many can you find? What do you notice? Durer lived from 1471 -1528. There is a clue in the magic square as to when the picture was painted. Can you guess which year it was painted? 1514 (in centre bottom row) Image credits: Ad Meskens via Wikimedia Commons

Now you have a go at some 4 x 4 magic squares, which use the numbers 1 -16. The magic total is 34, as in the Durer painting. a) Use 2, 7, 8, 12, 13 & 14 to fill in the square 4 9 6 15 b) Use 1, 2, 5, 6, 11, 12, 15 & 16 to fill in the square 9 1 11 5 16 3 10 7 4 3 10 13 8 1 14 When you have done both problems, try to make one up of your own. How many different 4 x 4 magic squares that use the numbers 1 -16 do you think there are?

Answer to first question: The magic total is 34, as in the Durer painting. 4 9 6 15 14 7 12 1 11 2 13 8 5 16 3 10 Use: 2, 7, 8, 12, 13, 14

Answer to second question: 9 6 15 4 9 16 5 4 16 3 10 5 6 3 10 15 2 13 8 11 12 13 8 1 7 12 1 14 7 2 11 14 Use 1, 2, 5, 6, 11, 12, 15 & 16 How many different 4 x 4 magic squares that use the numbers 1 -16 do you think there are? 880

The Passion Façade of the Sagrada Familia, Barcelona Image credits: Jeremy Keith, Bernard Gagnon - all via Wikimedia Commons

The Passion Façade of the Sagrada Familia, Barcelona • What do you notice about this magic square? • What is its magic total? 33 • How is it different to the Durer one? • How is it the same as the Durer one? Image credits: Jeremy Keith, Bernard Gagnon - all via Wikimedia Commons

Take Durer’s magic square Turn it through 90° again Compare the two magic squares 1 Image credits: Ad Meskens via Wikimedia Commons 14 15 4 12 7 6 9 8 11 10 5 13 2 3 16 (Easy to read version of upside down Durer square)

Take Durer’s magic square Turn it through 90° again Essentially they’re the Compare the two magic squares same!! 1 Image credits: Ad Meskens via Wikimedia Commons 14 15 4 12 7 6 9 8 11 10 5 13 2 3 16 (Easy to read version of upside down Durer square)

Random Total Magic Square https: //www. youtube. com/watch? v=a. Qx. Cnmhq. Zko http: //www. numberphile. com/videos/magic_square_trick. html

Random Total Magic Square…. how is this “magic” done? n-20 n-21 n-18 n-19

A Special Date Magic Square a b c d (i) Place the special date in the first row e f g h i j k l (ii) b+c = m+p [there are many different possible values] m n o p (iii) a+p = g+j [there are many different possible values] (iv) m+d = f+k [there are many different possible values] (v) b+n = g+k (vi) c+o = f+j (vii) a+m = h+l [there are many different possible values] (viii) All rows, columns and diagonals must add up to the same total, so e & i are determined https: //nrich. maths. org/1380

We hope you have enjoyed exploring magic squares with us! What questions do you have? Any unanswered questions can be written down and emailed to “Ask the Ri Masterclass Team” using this email masterclasses@ri. ac. uk We don’t know all the answers instantly, but we will find out and get back to you before the next Masterclass. Any comments you have about what you enjoyed or what you’d like to do more of can be written on the post-it note and handed in.

What else can I do to extend my knowledge of magic squares? ? https: //nrich. maths. org/6215 Different magic square https: //nrich. maths. org/87 Magic constants https: //nrich. maths. org/1205 Domino magic rectangle Try these as extra activities in class, or try them at home…