Highest and Lowest

Ashkan from Gorsefield Primary used the numbers 3, 4, 5 and 6, and tried not to use an operation sign more than once in each solution. They explained:

The highest I found was:

3 - 4 + (5 × 6) = 29.

I did 5 × 6 which = 30. Then I did 3 - 4 which is -1. Finally I did -1 + 30 = 29.

The lowest I found was:

3 - 4 + (5 / 6) = 0.167

I did 5 / 6 = 0.833. Then I did 3 - 4 = -1. Finally I did -1 + 0.833 = 0.167.

Kestrel class at Churchfields, the Village School in Wiltshire also used the numbers 3, 4, 5 and 6, and they experimented with using the same operations signs and different operations signs:

We decided that we had to keep the numbers in order.

At first we could use the operations as many times as we wanted.

Our highest score was 360

3 × 4 × 5 × 6 = 360

Our lowest score was -117.

3 – (4 × 5 × 6) = -117

At first we experimented with division before we realised that negative numbers were lower.

Then we decided to see what would happen if we could only use each symbol once (but we kept the numbers in order).

Our highest score was 34.5.

(3 ÷ 4 + 5) × 6 = 34.5

Our lowest score was -51.

3 – ((4 + 5) × 6) = - 51

Thank you, Kestrel class. What a good idea to decide on particular 'rules', like having the numbers in order, or using each operation once.

Brodie and Tully from St Patrick's School Macksville NSW in Australia looked at the numbers 3 to 6 and then at the numbers 1 to 6:

Highest with 3 to 6:

(4-3+5)×6=36

Lowest with 3 to 6:

6/3+4-5=1

(4-3+5)/6=1

Highest with 1 to 6:

(3+4)/(2-1)×5×6=210

Lowest with 1 to 6:

(4-3)×2/1+(5-6)=1

M, T and G from Monteney Primary School, Sheffield looked at the numbers from 1 to 6, and just used the multiplication symbol to find the highest possible number. They explained:

You need to multiply all the numbers together but not one. Multiplying all of the numbers 2, 3, 4, 5 and 6 together generate a total of 720.

2 × 3 = 6

6 × 4  = 24

24 × 5 = 120

120 × 6 = 720

If you add one to your total you get a total of 721.

720 + 1 = 721

You add one to make the biggest total, because if you multiplied your total by one then your answer of 720 wouldn't change.

The children at Holmwood House School in Colchester, Essex had a similar idea, but they did something different instead of adding the 1 at the end:

For the highest, we worked out that multiplying was the best way to enlarge the numbers, so we did 6 × 5 × 3, which came to 90. Multiplying that by 4 made 360. We then added the 2 and the 1 to make another 3, which we then multiplied by the 360 to make 1,080.

This can be written as 6 × 5 × 3 × 4 × (2 + 1) = 1080, which is the highest number anybody has found so far!

They looked for the lowest possible number by going into negative numbers:

For the lowest, we subtracted 3 from 1 to make -2, before multiplying the other remaining numbers by -2 to make an even larger negative number, using our knowledge of multiplying negatives and positives together. This ended up as -2 × 2 = -4, then -4 × 4 = -16, then -16 × 5 = -80, then finally, -80 × 6 = -480.

This can be written as (1 - 3) × 2 × 4 × 5 × 6 = -480. This is the lowest number anybody has found so far, but it is possible to make an even lower number using some similar ideas. If you think you might have found a lower number, email us to let us know.