A beginner's guide to five-number, three-digit-target questions

Introduction

For five-number questions, the target lies between 100 and 999. Finding a nearby starting value in the hundreds may feel unfamiliar at first.

This article collects essential formulas and techniques that are easy to remember at each level:

Level 1: Powers ( $x^n$ )
Level 2: Square roots ( $\sqrt{}$ )
Level 3: Factorials ( $n!$ ) and summation ( $\sum$ )

It also includes a reference table showing which formulas are useful for each target range.

Solving technique: starting value ± remaining numbers
Reference: formulas by target range
Level 1: powers
Level 2: square roots
Level 3: factorials
Level 3: summation
Basic: addition, subtraction, multiplication, and division
Frequently asked questions

Solving technique: starting value ± remaining numbers

The approach:

As soon as you see the target, look for a familiar value close to it.
Try to construct that starting value using some of the given numbers.
Add or subtract the remaining numbers to adjust it toward the target.

Example: Numbers 12678, target 727

The target 727 is close to 729 ( $3^6$ ), so we need to subtract 2.
To make $3^6$ , there is no 3 directly available. But $(8-6) = 2$ does not help… try again.
Use $(8-6) = 2$ as the base? Then the exponent needs to be 9, which is not available.
Return to $3^6$ : we need 3 and 6. We have $(1+2) = 3$ , and 6 is already available!
That leaves 7 and 8. $(8-6) = 2$ ❌—try again: 7 and 8 give $(8-7) = 1$ , or $8-6 = 2$ .
Answer: $3^6 - (8-6) = 729 - 2 = 727$ ✓

Reference: formulas by target range

Range	Powers	Factorials	Summation (basics)
100-150	125, 128	120, 140, 144	105, 110, 120, 136, 140
150-250	216, 243	168, 180, 210, 240	156, 171, 204, 210, 231, 240
250-400	256, 343	252, 280, 315, 336, 360	272, 285, 300, 325, 342, 378, 385
400-600	512	420, 480, 504, 576, 600	385, 406, 420, 462, 465, 528
600-800	625, 729	630, 720	600, 630, 650, 666, 718, 719, 744, 756
800-999	—	840, 960	829, 840, 849, 851, 873, 879, 894, 903, 930

The sections below show how to obtain each value.

Level 1: powers ( $x^n$ )

Main technique: Use a power to create a value close to the target.

Starting values to remember

Starting value	Formula
125	$5^3$
128	$2^7$
216	$6^3$
243	$3^5$
256	$2^8$ , $4^4$
343	$7^3$
512	$2^9$ , $8^3$
625	$5^4$
729	$3^6$ , $9^3$

Powers table

Base	$^2$	$^3$	$^4$	$^5$	$^6$	$^7$	$^8$	$^9$	$^{10}$
2	4	8	16	32	64	128	256	512	1024
3	9	27	81	243	729	-	-	-	-
4	16	64	256	1024	-	-	-	-	-
5	25	125	625	-	-	-	-	-	-
6	36	216	1296	-	-	-	-	-	-
7	49	343	-	-	-	-	-	-	-
8	64	512	-	-	-	-	-	-	-
9	81	729	-	-	-	-	-	-	-
10	100	1000	-	-	-	-	-	-	-
11	121	1331	-	-	-	-	-	-	-
12	144	-	-	-	-	-	-	-	-
13	169	-	-	-	-	-	-	-	-
14	196	-	-	-	-	-	-	-	-
15	225	-	-	-	-	-	-	-	-

Squares ( $x^2$ )

Values of $x^2$ between 100 and 999:

$x$	$x^2$	$x$	$x^2$	$x$	$x^2$
10	100	18	324	26	676
11	121	19	361	27	729
12	144	20	400	28	784
13	169	21	441	29	841
14	196	22	484	30	900
15	225	23	529	31	961
16	256	24	576
17	289	25	625

Example questions

Numbers	Target	Equation
02568	513	$6^0 + 8^{5-2} = 1 + 512 = 513$
35613	181	$\frac{3^6 - 5}{1+3} = \frac{724}{4} = 181$
57037	378	$5 \times 7 + 7^3 = 35 + 343 = 378$
23379	907	$(3 + 3 \times 9)^2 + 7 = 30^2 + 7 = 907$

Level 2: square roots ( $\sqrt{}$ )

Main technique: Use square roots to create additional numbers for bases or exponents.

Frequently used square roots

Formula	Value	Use
$\sqrt{4}$	2	Base $2^n$ , Exponent $x^2$
$\sqrt{9}$	3	Base $3^n$ , Exponent $x^3$
$\sqrt{16}$	4	Exponent $x^4$
$\sqrt{25}$	5	$5! = 120$
$\sqrt{36}$	6	$6! = 720$

More starting values to remember

Starting value	Formula	Example
128	$(\sqrt{4})^7$	$(\sqrt{4})^7 + 3 = 131$
243	$(\sqrt{9})^5$	$(\sqrt{9})^5 - 2 = 241$
512	$(\sqrt{4})^9$	$(\sqrt{4})^9 + 6 = 518$
729	$(\sqrt{9})^6$	$(\sqrt{9})^6 - 4 = 725$

Example questions

Numbers	Target	Equation
14847	123	$(\sqrt{4})^{8-1} - (7-\sqrt{4}) = 2^7 - 5 = 123$
13169	732	$(1+3) + (\sqrt{9})^6 - 1 = 4 + 729 - 1 = 732$
24479	512	$(\sqrt{4})^9 + (7-4-\sqrt{4}) = 512 + 0 = 512$
23469	185	$2 + \sqrt{9} \times ((\sqrt{4})^6 - 3) = 2 + 3 \times 61 = 185$

Level 3: factorials ( $n!$ )

Main technique: Use factorials to build a value near the target. This is especially useful for five-number questions!

Starting values to remember

100-150	150-250	250-400	400-600	600-800	800-999
$5!=120$	$4! \times 7=168$	$7!/20=252$	$7!/12=420$	$7!/8=630$	$5!+6!=840$
$7!/36=140$	$6!/4=180$	$7!/18=280$	$5! \times 4=480$	$6!=720$	$5! \times 8=960$
$3! \times 4!=144$	$7!/24=210$	$7!/16=315$	$7!/10=504$		$5! \times 7=840$
	$6!/3=240$	$7!/15=336$	$7!/9=560$		$7!/6=840$
	$5! \times 2=240$	$6!/2=360$	$4! \times 4!=576$
		$7!/14=360$	$5! \times 5=600$

Example questions

Numbers	Target	Equation
01449	125	$(9-4)! + (1-0) \times 5 = 120 + 5 = 125$
02346	720	$(2+4)! + 0 \times 3 = 6! = 720$
14478	728	$(7-1)! + (4+4) = 720 + 8 = 728$
11257	841	$5! \times 7 + 1 \times 1 = 840 + 1 = 841$
36025	608	$3^6 - (5! + 2^0) = 729 - 121 = 608$

Level 3: summation ( $\sum$ )

Main technique: Try summation when factorials and powers do not get you close enough to the target.

$\sum i$ : adding from 1 to n

Common upper bounds include 14 ( $2 \times 7$ ), 18 ( $3 \times 6$ ), 20 ( $4 \times 5$ ), 21 ( $3 \times 7$ ), 24 ( $4 \times 6$ ), 25 ( $5 \times 5$ ), 28 ( $4 \times 7$ ), 30 ( $5 \times 6$ ), 35 ( $5 \times 7$ ), 36 ( $6 \times 6$ ), and 42 ( $6 \times 7$ ).

100-150	150-250	250-400	400-600	600-800	800-999
$\sum_{i=1}^{2 \times 7}i=105$	$\sum_{i=1}^{3 \times 6}i=171$	$\sum_{i=1}^{4 \times 6}i=300$	$\sum_{i=1}^{4 \times 7}i=406$	$\sum_{i=1}^{5 \times 7}i=630$	$\sum_{i=1}^{6 \times 7}i=903$
$\sum_{i=1}^{3 \times 5}i=120$	$\sum_{i=1}^{4 \times 5}i=210$	$\sum_{i=1}^{5 \times 5}i=325$	$\sum_{i=1}^{5 \times 6}i=465$	$\sum_{i=1}^{6 \times 6}i=666$
$\sum_{i=1}^{4 \times 4}i=136$	$\sum_{i=1}^{3 \times 7}i=231$	$\sum_{i=1}^{3 \times 9}i=378$	$\sum_{i=1}^{4 \times 8}i=528$

$\sum (i+i)$ : summing 2i

100-150	150-250	250-400	400-600	600-800	800-999
$\sum_{i=1}^{2 \times 5}(i+i)=110$	$\sum_{i=1}^{3 \times 4}(i+i)=156$	$\sum_{i=1}^{4 \times 4}(i+i)=272$	$\sum_{i=1}^{4 \times 5}(i+i)=420$	$\sum_{i=1}^{4 \times 6}(i+i)=600$	$\sum_{i=1}^{4 \times 7}(i+i)=812$
	$\sum_{i=1}^{2 \times 7}(i+i)=210$	$\sum_{i=1}^{3 \times 6}(i+i)=342$	$\sum_{i=1}^{3 \times 7}(i+i)=462$	$\sum_{i=1}^{5 \times 5}(i+i)=650$	$\sum_{i=1}^{5 \times 6}(i+i)=930$
	$\sum_{i=1}^{3 \times 5}(i+i)=240$			$\sum_{i=1}^{3 \times 9}(i+i)=756$

$\sum (i \times i)$ : summing squares

100-150	150-250	250-400	400-600	600-800	800-999
$\sum_{i=1}^{7}(i \times i)=140$	$\sum_{i=1}^{8}(i \times i)=204$	$\sum_{i=1}^{9}(i \times i)=285$	$\sum_{i=1}^{2 \times 5}(i \times i)=385$	$\sum_{i=1}^{3 \times 4}(i \times i)=650$

Summation with factorials

Common forms:

Summand	Bounds	Result	Notes
$i + i!$	(5, 6)	851	Very common
$i + i!$	(4, 6)	879
$i + i!$	(1, 6)	894
$i \times i!$	(1, 5)	719
$i \times i!$	(2, 5)	718
$i! - i$	(5, 6)	829
$i! - i$	(4, 6)	849
$(i+i)!$	(1, 3)	746	$=(2!)+(4!)+(6!)=2+24+720$
$(i+i)!$	(2, 3)	744	$=(4!)+(6!)=24+720$
$i!$	(1, 6)	873
$i!$	(5, 6)	840	$=5!+6!=120+720$

Example questions

Numbers	Target	Equation
12568	465	$\sum_{i=1}^{5 \times 6} i + (8-6-2 \times 1) = 465 + 0 = 465$
05577	650	$\sum_{i=1}^{5 \times 5} (i+i) + (7-7) \times 0 = 650$
33669	669	$\sum_{i=1}^{6 \times 6} i + (9-3-3) = 666 + 3 = 669$
13578	629	$\sum_{i=1}^{5 \times 7} i - (8-3-1-3) = 630 - 1 = 629$
24566	851	$\sum_{i=5}^{6} (i+i!) + (4-2) \times (6-6) = 851 + 0 = 851$

Basic: addition, subtraction, multiplication, and division

Basic level uses only $+$ , $-$ , $\times$ , and $\div$ . There is no fixed formula—you need to practice finding combinations yourself!

What to try first if you are unsure of the level

Start with factorials, especially if the question includes 5, 6, or 9. Look for ways to make 5! or 6!.
Try powers if the target is close to 128, 256, 512, or 729.
Try summation for distinctive targets such as 300, 465, or 666.
Return to Basic operations if nothing else works, and look for suitable factors.

Frequently asked questions

What does “level” mean in Fastmath?

Level	Allowed operators
Basic	$+$ , $-$ , $\times$ , $\div$
Level 1	Basic + Powers
Level 2	Level 1 + roots (no nesting)
Level 3	Level 2 + Factorials, summation (roots may be nested two levels deep)

General rules:

The result must be a positive integer.
Use every number provided in the question.
Do not concatenate digits to make decimals or multi-digit numbers.

What is sigma notation?

Sigma, or summation, adds terms from a starting value through an ending value.

For example:

$\sum_{i=1}^{4}i = 1 + 2 + 3 + 4 = 10$
$\sum_{i=2}^{3}i^2 = 2^2 + 3^2 = 4 + 9 = 13$
$\sum_{i=2}^{3}i! = 2! + 3! = 2 + 6 = 8$

Learn more in What is sigma notation?.

What is a factorial?

A factorial, written with $!$ , multiplies the integers from 1 through the given number.

For example:

$3! = 1 \times 2 \times 3 = 6$
$4! = 1 \times 2 \times 3 \times 4 = 24$
$5! = 1 \times 2 \times 3 \times 4 \times 5 = 120$
$6! = 720$

A special case is $0! = 1$ , by mathematical definition.

Learn more in What is a factorial?.

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A beginner's guide to five-number, three-digit-target questions

Introduction

Contents

Solving technique: starting value ± remaining numbers

Reference: formulas by target range

Level 1: powers ( $x^n$ )

Starting values to remember

Powers table

Squares ( $x^2$ )

Example questions

Level 2: square roots ( $\sqrt{}$ )

Frequently used square roots

More starting values to remember

Example questions

Level 3: factorials ( $n!$ )

Starting values to remember

Example questions

Level 3: summation ( $\sum$ )

$\sum i$ : adding from 1 to n

$\sum (i+i)$ : summing 2i

$\sum (i \times i)$ : summing squares

Summation with factorials

Example questions

Basic: addition, subtraction, multiplication, and division

What to try first if you are unsure of the level

Frequently asked questions

What does “level” mean in Fastmath?

What is sigma notation?

What is a factorial?

More articles

The order of operations

A review of Fastmath.io

What is a factorial?

What is exponentiation?

$x$	$x^2$	$x$	$x^2$	$x$	$x^2$
10	100	18	324	26	676
11	121	19	361	27	729
12	144	20	400	28	784
13	169	21	441	29	841
14	196	22	484	30	900
15	225	23	529	31	961
16	256	24	576
17	289	25	625

$x$	$x^2$	$x$	$x^2$	$x$	$x^2$
10	100	18	324	26	676
11	121	19	361	27	729
12	144	20	400	28	784
13	169	21	441	29	841
14	196	22	484	30	900
15	225	23	529	31	961
16	256	24	576
17	289	25	625

Introduction

Contents

Solving technique: starting value ± remaining numbers

Reference: formulas by target range

Level 1: powers (xnx^nxn)

Starting values to remember

Powers table

Squares (x2x^2x2)

Example questions

Level 2: square roots (\sqrt{}​)

Frequently used square roots

More starting values to remember

Example questions

Level 3: factorials (n!n!n!)

Starting values to remember

Example questions

Level 3: summation (∑\sum∑)

∑i\sum i∑i: adding from 1 to n

∑(i+i)\sum (i+i)∑(i+i): summing 2i

∑(i×i)\sum (i \times i)∑(i×i): summing squares

Summation with factorials

Example questions

Basic: addition, subtraction, multiplication, and division

What to try first if you are unsure of the level

Frequently asked questions

What does “level” mean in Fastmath?

What is sigma notation?

What is a factorial?

More articles

The order of operations

A review of Fastmath.io

What is a factorial?

What is exponentiation?

Level 1: powers ( $x^n$ )

Squares ( $x^2$ )

Level 2: square roots ( $\sqrt{}$ )

Level 3: factorials ( $n!$ )

Level 3: summation ( $\sum$ )

$\sum i$ : adding from 1 to n

$\sum (i+i)$ : summing 2i

$\sum (i \times i)$ : summing squares

$x$	$x^2$	$x$	$x^2$	$x$	$x^2$
10	100	18	324	26	676
11	121	19	361	27	729
12	144	20	400	28	784
13	169	21	441	29	841
14	196	22	484	30	900
15	225	23	529	31	961
16	256	24	576
17	289	25	625