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Change the numbers on the apples so that the number on the lemon is 88.

[Double click an apple number to change it]

The Lemon Law states that the numbers on the apples are all single digit positive integers. The numbers on the oranges combine the apple numbers as tens and units. The number on the lemon is the sum of the oranges.

Can 88 be made according to the lemon law?
If not, can you prove it?
If it can be, how many different ways can it be done?

Which numbers can be made by the lemon law?

## A Mathematics Lesson Starter Of The Day

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Topics: Starter | Algebra | Functions | Puzzles

• Liam Kelsall, Holden Lane High
•
• K Hudson, Drigg
•
• Elegant starter with lots of fun ideas to explore. The kids soon arrived at 2222 and went from there!
• Akbar Ali, Whitefield Community School
•
• Both 2222 and 1331 give 88. This is because 22+22+22+22=88 and 13+31+13+31=88 I believe these are the only two solutions due to the symmetry in the answers.
• Chelsea Maher, Harrytown High School Stockport
•
• without being told the single digit integer ruler, Chelsea Maher came up with

1 2
3 6.5

Well done
• Mr Wilson, 2MW
•
• Our Year 5 class, (all of whom are brainboxes obviously!) have come up with 2222, 3113, 4004, 1421 and 3023
• J Nicholl, Kellett School Hong Kong
•
• A pupil in my class saw the pattern that the apple digits must add up to 8.
• Miss Hickman, Wildern
•
• My year nines found that the combination 1241 was a solution not already listed
• Mrs Simpson and 4M4, Torry Academy, Aberdeen
•
• David in 4M4 found this solution: 0, 0, 8, 0
• Yeah!, Catherine Potter
•
• It was really fun! All my family and I did it.
• Mr Baker, Somerset, UK
•
• My year 8 class loved this starter
Soon found that if apples were lettered
a b
c d
then 2a+b+c=8 and b+c+2d=8
Brilliant one thanks :).
•
• 0,5,3,0 and the mirror also work.
• Mr Peter Okurut, Benjamin Britten High School, Lowestoft
•
• Using algebra,
40
04.
• Mr Page & 7J1, John Ferneley College
•
• 7J1 found out using a,b,c,d to stand for the apples that:-
a + b + c + d = 8
a = d (Archie Herrick & Josh Leach).
• Mrs Gocht, Year 3
•
• Also 1241 works, my class loved it and I have year 3, but it did take them awhile! bless there little hearts!
• Alistair Carratt, Sanderson High School
•
• I work in an ASN school. My S5B class tried this puzzle and one pupil found 2222 and another found 1331. This led us to find a pattern - all numbers must add up to 8 and the grid must have symmetry across the diagonals. We found 25 solutions.
• Mrs Ward, Cleeve School
•
• We found out that 06
20 is a possible of 88.
• 6T, Widford Lodge
•
• We found 1511 and 1601 too. It was good fun!
• Mrodwyer, Churchdown School Glos Year 8
•
• Year 8 pupils did this 5 minute starter and arrived at all the previous answers already posted. very sharp class. and would like to add 0800.
• Mrs. W And 1C1, Torry Academy, Scotland
•
• We found, 2222 by Donnelly and 1331 by Natasha then noticed they added to 8. Great!
• Year 6M, Garswood Primary
•
• We tried this in staff meeting verses our year sixes. Year Six got the answer quicker than some of our teachers!
• Mr J Saye, 8
•
• Castle Rock High School Coalville
My Set 1 Year 8 class had 5 minutes to complete this task, one girl came up with another way to get 88 in the lemon:
2312
Well done to RMc!!!
Mr Saye.
•
• Lemon Number 95
20
50
also works.
• Roy Froud, Bournemouth
•
• I would like to ask students if they could solve the problem sent home from school to my 9-year-old great nephew:
a b
c d

In the grid shown, replace letters a, b, c, d, by 4 different numbers from 1 to 9 so that adding the two rows and two columns together sums to 200.
Example:
7 1
3 8

71 + 38 + 73 + 18 = 200
See if you can find the 10 other combinations that work
In all the above, b and c are interchangeable, leading to another 11 solutions, although, as has been pointed out to me by my nephew, this amounts to transposing rows and columns.
For fun, prove that d = a +1 and that a + b + c + d = 19
Roy Froud
Voluntary Maths Tutor.
• Lars, Norway
•
• Why does the answers not show any solutions where an apple has the number 0?
I used this way to solve the problem:
The apples are labeled:
a b
c d
the oranges are
e = 10*a + 1*b
f = 10*c + 1*d
g = 10*a + 1*c
h = 10*b + 1*d
the lemon is then the toal sum:
lemon = e + f + g + h
= 20*a + 11*(b + c) + 2*d (after doing some algebra)
Then it's easy to see alle the solutions which adds up to 88. you can start with saying that we put a=0, this means that to get 8*10, we must have that b+c=8. In this case, the lemon will always be 88, and we must put d=0. This means that every combination of:
a, d = 0 and b + c = 8
will give 88. There are 9 different combinations of this.
Next solution can be found by setting a=1. we then need to add tens up to 80, which can only be done if b+c = 6. If so, d must be equal to 1. This gives:
a, d = 1 and b + c = 6
Every combination of this gives a lemon with 88, there are 7 different combinations
In fact, we must always have that a = d. There rest of the solutions are:
a, d = 2 and b + c = 4. This solution has 5 different combinations
a, d = 3 and b + c = 2. This solution has 3 different combinations
a, d = 4 and b + c = 0. This solution has only one combinations.
In totale, there are therefore 25 different ways to get the lemon number 88.
• Ella, St.James CEC Primary
•
• Me and Mr.Wealend all came up with 1331 but Mr.Wealend said we had to get 88
and the apples can't add up to 8. We came up with 4004 and 2113 but they add up to 8.We sill can't get it.I found out that the apples had to go together to a two digit number which comes up in the orange then add together to get the number in the lemon.
•

• Sarah, St James Ptimary
•
• 40
04
Is a answer but we are trying to do it without it adding up to 8. It is a very hard problem but we like to do challenging problems.
• Ben L, St James' Primary
•
• I found that the numbers in the apples have to add up to 8 and if you can find one that does not add up to 8.........your very clever.
• Sarah And Molly, Mr Waelend's Maths Group
•
• It doesn't work unless the number add up to 8. We tried putting 1 in all the boxes, and then added one to it until it made 88. The one that we got up to in the end was 16 Our headteacher set us a challenge of finding 88
01.
without the numbers in the apples adding up to 8.We think it is impossible. We will keep trying though.
• Ben, St James
•
• I found the 1313 solution quite quickly and then found others like 2222 and 4004 however we rialised that all these combination's digits add up to 8 so to try and discover if you can do it where the digits don't add up to 8 we worked out what happened when we changed each number:
top left: +/- 20 top right:+/-11
bottom left:+/-11 bottom right:+/-2
Once I had found this information I tried to make 88 by multiplying these different numbers by digits that when added don't make 8 I couldn't find any without going in to decimals (which the thing dosn't allow).
• Alex+Conall, Mr Waelends Maths Group
•
• We started off by doing 3131 then we started trying to do nunbers that didn't equal 8 and worked out that you needed to make numbers that made the numbers in the oranges equal 88.
• Mr W, St James Ce School
•
• We found the answer 1313 very quickly and felt extremely pleased with ourselves, so we set ourselves the challenge of trying to find other solutions. We found a number of them and discovered that the total on the apples always seemed to come to 8.
We set out to try and find a solution that didn't add up to 8. In order to do this, we tried to find out the effect of each apple on the number on the lemon.
In doing this, we soon discovered that the apple in the top left hand corner increased the lemon by 20 when it was increased by one. We could see that this was because it was used as the tens digit twice in making the lemon number. The bottom right hand corner apple was only used as the units digit twice so added 2 to the lemon when the number in it was increased by 1. The remining two apples were used once as a ten and once as a unit, so they added 11 to the lemon when they were increased by 1.
Suddenly the problem becomes much simpler. How can we make 88 by adding 20s, 11s and 2s?
We felt a little frustrated that we couldn't use numbers larger than 9 or decimals. If we could then we could simply put 44 in the bottom right hand corner. However, even 4+4 =8!
We were split in our minds about whether there was a solution or not.
I tried by starting with no apples in the TL. To reach 88 I could share 8 apples between the BL and TR apples and leave 0 in the BR apple. But this would add up to 8!
If I only spread 7 apples between the two corners it would leave me 11 to make up with the BR corner. I can't make 11 as I need an even number. Every number in the BR corner adds 2 to the lemon.
So I need to spread 6 apples between the TR and BL corners, leaving 22 to be added. I can only do this by adding 11 to the BR corner. Too many.
In carrying on in this way, we could find no combination of numbers that didn't add to 8!
• Rachel, St James' CEC Primary School
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• We found 2222 and 1313 like everyone else but, as we thought it was a bit too easy, we decided to find one that didn't add up to 8 so here is what we found:
I worked out that every time you change the top left apple by 1 (up or down) the lemon increases or decreases by 20. When you increase or decrease the bottom right apple by 1 (up or down) the lemon increases or decreases by 2. Then the top right apple, if changed by one (up or down) causes the lemon to decrease or increase by 11. Ditto for the bottom left apple.
I am pretty sure it is impossible unless you can use decimal points and double digit numbers.
• Matthew, 3D, Craigslea State School
•
• I found five possibilities;
40 31 22 04 13
04 13 22 40 31.
• Tc, Thailand
•
• I labelled a b c d
and then found that 88 = 2(10a + b) + 11(b + c)
so 88 = 66 + 22
or 88 = 44 + 44
or 88 = 22 + 66
see table below.. I got 13 combinations for a b c d
a d 10a + b b c b + c
1 3 3 33 1 1 2
2 2 2 22 2 2 4
3 2 2 22 1 3 4
4 2 2 22 3 1 4
5 2 2 22 4 0 4
6 2 2 22 0 4 4
7 1 1 11 0 6 6
8 1 1 11 6 0 6
9 1 1 11 1 5 6
10 1 1 11 5 1 6
11 1 1 11 2 4 6
12 1 1 11 4 2 6
13 1 1 11 3 3 6.
• 5HC, The Beacon
•
• This was fun!
Manhar started us off by figuring that one particular digit works in all four slots.
We ended up working out that there were 25 ways of doing it (and 50 if you allowed negative single-digit integers!). In between, Nayan spotted that the digits added up to 8, which really helped us, as did the realisation that the top right and bottom left apples always had the same sum.
• Roy Froud, Bournemouth
•
• To all who have answered my original question, it is implied by the separate letters a b c d that they are all different integers. so there are only the 11 or so answers I listed. Not sure about 0 as an integer. Not in my book but someone will argue!

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## Numbers and the Making of Us

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