Power Countdown

In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.

Age

14 to 16

Challenge level

Secondary

NUMBER

Powers and Roots

Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving

Being curious Being resourceful Being resilient Being collaborative

Power Countdown printable sheet

In the game of Power Countdown, you use a set of numbers to make a target number, but unlike the usual Countdown game where you can use $+, -, \times$ or $\div$, the only operations you can use are raising a number to a power, taking the reciprocal of a number, or finding the product of two numbers.

Each number can only be used once. You don't have to use all the numbers. There is often more than one way of making a particular target, so see how many different ways you can find.

Watch the video to see some examples.

Can you find any other ways of making $8$?

Are there any ways which use all the numbers?

Here is another selection.

How many ways are there to make the target number of $125$?

Below is a selection of numbers and five targets.

How many different ways can you find to make each target?

Are there any targets you can't make? How close can you get?

Other ways of making 8

Samuel from Verulam School in the UK: $\left(4^\frac12\right)^3$

Johnny from HKIS in Hong Kong: $\left(32^\frac15 \times 16^\frac14 \times \frac12\right)^3$

Tom from St Georges Church of England Academy: $\left(2^3\times16^\frac14\right)\div32^\frac15$

Ways of making 125 from 2, 4, 5, 25, 27, 81

Yuktha from Wallington High School for Girls in England: $25 \times 5 = 125$

Haneen from Ponteland High School in the UK and Yuktha both got this answer:

$5^{\left(81^\frac14\right)}=125$ because $81^\frac14 = 3$ and $5^3 = 125$

Numbers you can make using 2, 5, 16, 243, 343, 512

Niall from JAPS, Also Jai and Charlotte from Tytherington High School Macclesfield and Haren from Jebel Ali Primary School all got 1024 in the same way. Haren wrote:

16 to the power of 5 is 1,048,576

1,048,576 to the power of $\frac12$ is 1024

Equivalently, $16^\frac12=4$ and $4^5=1024$.

Jai and Charlotte made 64: $16^\frac12 = 4,$ $243^\frac15=3,$ $4^3 = 64$

Daniel, Alex, Nick and George from Tytherington High School also made 64:

$512^\frac13 =8$

$8^2$

Tom from Devonport Boys made 49:

$243 ^ \frac15 = 3$

$343 ^ \frac13 = 7$

$7 ^ 2= 49$

Daniel, Alex, Nick and George said it is impossible to make 89:

89 cannot get exactly as it's prime 81 is as close as we got.

Tom went into more detail.:

89 is impossible to achieve exactly:

$2=2^1$

$5=5^1$

$16=2^4$

$243=3^5$

$343=7^3$

$512=2^9$

all the numbers we are using can be expressed as a single prime to a power.
raising these numbers to any (pos. integer) power will only increase the value to which the prime has been raised.
roots of primes are irrational so rooting these numbers beyond the powers which are already there will only give irrational results: no use to us.
89 is prime but we don't have any numbers which we can use that have 89 as a factor and because we cannot multiply or divide there is no way of changing these factors to achieve a multiple of 89.
I believe a similar argument can be applied for 216 as 216=6$^3$ and we don't have [enough] usable numbers with 6 (or specifically 2 and 3) as a factor.

Why do this problem?

This problem provides a context for playing with integer and fractional powers so students can get a feel for manipulating indices and become more familiar with commonly used powers of smaller numbers.

Possible approach

You could start by showing the video in the problem. Alternatively, show students the first set of numbers $2, 3, 4, 5, 16, 32$ and the target of $8$, and set the challenge of making it using only powers, products and reciprocals. There is more than one way of making $8$, so students could be challenged to find as many different ways as possible, or to look for a way that uses all the numbers.

Once everyone is confident with the concept of using powers and roots to make the target, make a start on the second challenge to make $125$. Students could work in pairs, discussing how to write the target and each of the starting numbers in terms of powers and roots, and writing solutions in terms of fractional powers. In answering the question "How many ways are there?" there is the opportunity for discussion about the order of operations, and why cube rooting followed by squaring is the same as squaring followed by cube rooting, for example.

The last part of the problem is about recognising commonly used powers. The key to finding an answer to those targets which are possible is to spot how to write the target in terms of powers. Students could work on one particular target and then feed back to the rest of the class a solution, or an explanation of how they know their target was impossible.

They could follow this up by making targets of their own and challenging each other to find a way of getting the target.

Key questions

Why is raising to a fractional power the same as finding a root?

Can you spot any numbers which can be written as another number raised to a power?

Why are some totals impossible?

Possible support

Give students a selection of numbers and targets to work with which are less than $100$, concentrating mainly on powers of $2$ and $3$.

Possible extension

Students could make up sets of numbers and targets for each other, using larger powers, and write answers using fractional powers with numerators other than $1$.

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