Is it possible to rearrange the numbers, row and column headings so that this table is mathematically correct? If not, how many numbers can you place?

A Mathematics Lesson Starter Of The Day

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link to the interactive, self-checking pupil version of

Topics: Starter | Number

  • Cmurtaugh, Park High School
  • This did not work can not have an even and odd number
  • Peter Mayne, Crieff High School
  • This puzzle is impossible! There are not 4 multiples of 5 from 1 to 16.
  • Year 5 , Ardingly College
  • We have also found this impossible to do!!
  • A Dixon, NDTC Doncaster
  • If you continue to read down the page it does say that it is impossible but asks for pupils to prove why!
  • Daniel, Honk Kong
  • No wonder everyone thinks its impossible! It said to rearrange everything! Not just the numbers.
  • 8x2, Portchester Community School
  • This is impossible as the headings do not make sense.
  • Scott, Birmingham
  • Surely you also need another multiple of 5?
  • Year 8, Wyedean
  • This is too hard for my year 8 class, is it possible?
  • Mrs Humphreys' maths group, Nevill Road, Bramhall, Stockport
  • The children were frustrated that it didn't work but it generated some good discussion.
  • C Southward, Carlisle
  • My Year sevens were amazed at some of the earlier comments, obviously these people had not even understood the question.
  • J.Barclay-Devine, Ashton, Bedfordshire
  • Once my year 6 class decided that some properties would not go together we tried to fill in the grid using different numbers. The problem generates a lot of discussion. Presumably this is what the comment meant when saying people hadn't understood.
  • Kerry Meaden-Kendrick, Bangkok Patana
  • My Year 5 maths class found this VERY frustrating-we were missing the fourth multiple of five!!!

    This cannot be done without it, surely???!!!

    We did well but no biscuit...:(
  • Year 8, set 1, Okehampton College, Devon
  • We don't think there is any way to arrange the column and row headings, even if you allow, say, 6 headings across the top and 2 headings down the side. This is because odd and even have to go in the same area (either row or column), because otherwise they will contradict each other. Also multiples of 4 have to go with these two, as multiples of 4 cannnot be odd, then prime numbers have to go with these, as primes cannot be multiples of 4. Then we saw that square numbers cannot be prime, so that's 5 headings together already. There are similar contradictions for all the headings except multiples of 3!
    There are not enough multiples of 3 to fit with all the other headings.

    Even then, there are not enough numbers which are multiples of 5!
  • Mrs A Morris, VCC Rutland
  • We found similar problems but it did cause a huge debate and uproar!
  • Ms. Toofail, Willowfield school Walthamstow
  • A student from my class got 11 of the numbers into the grid. Try and see if anyone can beat that, if so leave a comment.
  • Class 8a5, The Whitby High School, Ellesmere Port
  • Impossible! There are no numbers there that are both prime and square, odd and even and prime and multiple of 4.
  • Mr Barnett, Glenwood High School, Fife, Scotland
  • When the pupils rearranged the grid into a 5 x 3 grid, narrowing the options to 15, one of our pupils managed to fit in 12 numbers having managed the 11 numbers with the 4 x 4 grid.
    A great activity that we've spent a lot of time discussing
  • Mr Barnett, Glenwood High School (Again)
  • One of my 1st year girls has just brought a homework in with the 4 x 4 grid. We have all checked very carefully and she managed to fit 12 numbers into the grid.........we'll keep trying.
  • Mrs. Schultz, Hylands School, Chelmsford, Essex.
  • My year 8 class took 5 minutes to work this out by looking at the multiples of 5.
  • Emma Cox, Sir James Smiths School Camelford
  • My year 7 group worked really well on this. 4 students got 12 numbers in their grids and one took it home for a parent to get 13! Ive not seen this solution. They cut out numbers and headings to make manipulation easier.
  • Mr Jan, Haydock Sports College
  • Closest we got was 4 numbers left over! anyone beat that?
  • Hamish,
  • It clearly says at the bottom that its supposed to be impossible so why are so many people complaining that it is! don't use it then.
  • Yr 5/6 Class, Holy Eucharist , East Malvern. Melbourne.VIC. Australia
  • The problem was really hard and it created alot of discussion in our class.We couldn't solve it even when we concluded that "1" is a multiple of 5.It was frustrating when we could solve 3 numbers in either a row or column but we couldn't solve the other number.Some people didn't realise that you could move the titles as well.
  • Mrs Taylor's P7 Set, Carolside Primary
  • Although we found this a bit frustrating, we did investigate triangular numbers.
  • Yr 8, Sana Maria College
  • We found we could put all numbers except the number 14 onto the grid once we arranged the headings. Thanks to Emily.

  • Year 5, O, N, J, L, St Nicholas
  • We managed 12! Can anyone beat that?
  • Daniel, Maidenhead
  • My daughter asked me for help with this exercise and as it appeared to be impossible I believe that my (and most other people's) initial understanding of the problem was incorrect. Keeping this in mind I took a different approach and took a careful look at the row and column headings. Take one heading as an example 'Multiple of four', it is singular and not plural. Therefore I believe that the headings are not related to the cells but to the sum of the cells in each row and column. Taking this approach it is possible to complete the exercise. Have fun trying this out.
  • Year Six, Tickhill St. Mary's
  • Our Year Six class managed a maximum of 13 numbers placed. No-one thought to rearrange the grid to try to make it easier! Some thinking skills need developing...
  • Mr Bennett, St Stephen's C.E. Junior School
  • My clever Year 6 class found that this doesn't work. As a suggestion, they said: " make sure you can do it yourself before putting it on line!"
    " Keep the column and row titles but change the numbers?".
  • Miss Farmer + 8LM4, Landau Forte College, Derby
  • My year 8 group found this quite annoying! We tried for ages to rearrange things only to find there were not enough multiples of 5! good discussion though!
  • 8 Larch, St Anne's
  • St Anne's year eight did not like having a task that could not be done.
  • Mrs. Fischl, Grade 11 College Class
  • My class would like to see more of this, however, we would like a puzzle that can actually be solved. It would have been nice to have the instruction read, why is this puzzle impossible?
  • Miss James, Shiplake
  • This is impossible and my calss found it really Fustrating!
  • A Year 8 Student, Wilmslow High School, Wilmslow, Cheshire
  • It says at the bottom of the page that is impossible.
  • Emma Cox, Sir James Smiths School Camelford
  • Sam and Jared from my year 7 Maths group, managed to fit 13 numbers onto the grid by cutting the heading and numbers out to rearrange them. They quickly realised there were only 3 multiples of 5. Results were then put on a poster.
  • Hanxiao, Jinchengxiaoqu
  • Impossible!
    Because the multiples of 5 is not enough.
  • Chelsay Sharlotte, TVA
  • My teacher should have checked that it worked before he gave it to us.
  • 8X1, Hadleigh High
  • We also managed to only find 12 to fit in.
  • Mr Winning's Class, OLHS
  • This puzzle would be better if it was not impossible.
  • Mrs Price, York
  • My class would like to think they could get all the numbers in the grid however it is an impossible problem.
  • Niole Jordan, Charing
  • This is easy. Did you know that you could move the headings? I am 9 and I even know that.
  • J Cousins, Oakfield School Academy
  • This would be an excellent starter, if it worked!!
  • 7a2 - Harton Technology College, South Shields
  • We managed to enter 13 numbers and remove one heading.
  • Tia, Weston
  • Disappointed in this one.
  • GMU 5-7, Pairc School, Isle Of Lewis
  • It's impossible. You can't get an odd and even number.
  • Ashley Black, Sgoil Ainart
  • You can not have an even and odd.
  • Mrs Cooper, Cornelius Vermuyden School, Canvey Island
  • What a wonderful thinking starter... I was disappointed that the student version that I was told did work did not. It was different to the solution given, having got several of the headings different than in the question. If this were corrected it would be fantastic!
    Thank you Transum Starter of the Day!

    TRANSUM: 'Mrs Cooper, the student version has a button for changing the headings. The student version is indeed possible!'
  • Our Lady Of Lourdes, Years 6 and 7
  • This is impossible my class were very disappointed they couldn't figure the answers.
  • Mrs S, Wilmslow
  • Disappointing for a morning starter.
    I hoped my children could learn about numbers while I did register.
    Could have spent a lesson on it after I realised it was impossible!!
    No more!
  • Class 201, Cork, Ireland
  • We found this fun but tricky. It took a long time and we eventually found out that it wasn't possible. We created our own table which worked- this one doesn't.

  • Heather, The Priory Academy LSST, England
  • The reason it's impossible with those numbers is the number 14, which only fits into one category, even numbers, when every number has to fit 2 categories.
  • E23, Brighton Primary
  • It truly is impossible.
  • Year 5, Middlemarch School
  • We got 12 numbers on the grid and we are only Year 5!
    It should be called UNSATISFACTION.
  • Maryam, Cardiff
  • I got 13!
  • The Mathematician, ISM
  • It is possible, you can move the heading boxes! Our teacher figured it out, that's why we could do it!
  • Samuel, Horcell Cof E School
  • You would have to get factor of 42 , Odd numbers, Even Numbers and factor and four in one row to make this puzzle even possibe.
  • Sophia, St Mary's Collage
  • It is very possible you have to look at the two numbers in 16 e.g 1 and 6, 1 is an odd number and 6 is an even.
  • Mr. Foxworth, Woking
  • This is a challenging task but is helpful to the children's understanding.
  • Terry, Horsell C Of E Junior School
  • This is a good discursive game.
  • Ms Banks, Oxgangs Primary, Edinburgh
  • This is impossible!! My P7 class found this very frustrating. At first they thought they could do it, but soon realised it wasn't possible.
    Good discussion about prime numbers, triangular numbers and factors though.
  • James, Y9
  • Imposible! even if you change the headers.
  • Mr. Mildwin, Wodonga Primary
  • Caused some great discussion, however couldn't find a possibility that matched all requirements.
  • Class Five, Steephill School, Fawkham, KENT
  • We saw that the Odd and Even headings had to be together and also Prime and Multiple of Four headings because they couldn't be opposite as an odd cannot be even and no multiple of four is prime.

    We put the four mentioned across the top and the others down the side. We managed to fill most boxes until we came to the "Multiple of Four" row when we noticed that no factors of forty two or triangular numbers that we had were in the four times table.

    We decided that as we could only fit fourteen of the numbers in the chart that the task was impossible
  • 1537, 1537
  • It was impossible.
  • Me,
  • You guys really need to read it, first off it is asking how many numbers can you get in? Of course it is impossible, it literately says that!! You can also just move around the headings.
  • Jayden Marshall, Heathfield Community College
  • I got 13 it was very diffuclut by the way you can move the titles.
  • Elena Bezoari, Twitter
  • Jonathan, Califonia
  • It is impossible but managed to get in 13 numbers.

    How to do 13:
    14 10 4 6
    12 16
    13 15 9 7
    11 3 2.

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
Click here to enter your comments.

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Laptops In Lessons

Teacher, do your students have access to computers such as tablets, iPads or Laptops?  This page was really designed for projection on a whiteboard but if you really want the students to have access to it here is a concise URL for a version of this page without the comments:

However it would be better to assign one of the student interactive activities below.

Laptops In Lessons

Here is the URL which will take them to a student version of this activity.

Student Activity


Curriculum Reference

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