Photo by Nick Hillier on Unsplash

Which pair of digits is the most useful?

Introduction

Have you ever wondered how many different answers you can make with just two digits?

This article examines all 54 digit pairs, from (0,1) to (9,9), using:

  • Addition, subtraction, multiplication, and division (+,−,×,÷+, -, \times, \div)
  • Exponentiation (xnx^n)
  • Roots (x\sqrt{x})
  • Factorials (n!n!)
  • Sigma summation (∑\sum)

The results are striking: some pairs produce only three values, while others produce as many as 81!

Contents

  • The top five at a glance
  • 1st: 4 and 9 (81 values)
  • 2nd: 3 and 4 (77 values)
  • 3rd: 2 and 9 (49 values)
  • 4th: 4 and 5 (47 values)
  • 5th: 0 and 9 (45 values)
  • Why are some pairs more useful?
  • Frequently asked questions

The top five at a glance

Rank Digit pair Values produced Useful technique
1 4, 9 81 4=2\sqrt{4}=2, 9=3\sqrt{9}=3 can be used in factorials
2 3, 4 77 3!=63!=6, 4=2\sqrt{4}=2, 4!=244!=24
3 2, 9 49 2 and 9=3\sqrt{9}=3
4 4, 5 47 4=2\sqrt{4}=2, 4!=244!=24, 5!=1205!=120
5 0, 9 45 0!=10!=1 and the versatility of 9

Notice that 4 or 9 appears in every pair in the ranking. Their square roots give useful whole numbers: 4=2\sqrt{4}=2 and 9=3\sqrt{9}=3.

1st: 4 and 9 (81 values)

This pair is the most versatile because 4=2\sqrt{4} = 2 and 9=3\sqrt{9} = 3 provide small numbers that can be used in further operations, particularly 3!=63! = 6 and 6!=7206! = 720.

Result Expression Result Expression Result Expression
0 ∑i=23(i−i)\sum_{i=2}^{3}(i-i) 29 ∑i=23ii!\sum_{i=2}^{3}\sqrt{i}^{i!} 220 ∑i=23(i!)i\sum_{i=2}^{3}(i!)^i
1 4−94 - \sqrt{9} 30 4!+(9)!4! + (\sqrt{9})! 241 ∑i=23(i!)!i\sum_{i=2}^{3}\frac{(i!)!}{i}
2 (9)!−4(\sqrt{9})! - 4 31 ∑i=23ii\sum_{i=2}^{3}i^i 264 ∑i=924i\sum_{i=9}^{24}i
3 (9)!÷4(\sqrt{9})! \div \sqrt{4} 33 4!+94! + 9 271 ∑i=49i2\sum_{i=4}^{9}i^2
4 (9)!−4(\sqrt{9})! - \sqrt{4} 36 4×94 \times 9 283 ∑i=34ii\sum_{i=3}^{4}i^i
5 9−49 - 4 37 ∑i=34(i+i!)\sum_{i=3}^{4}(i+i!) 284 ∑i=29i2\sum_{i=2}^{9}i^2
6 4×9\sqrt{4 \times 9} 39 ∑i=49i\sum_{i=4}^{9}i 285 ∑i=624i\sum_{i=6}^{24}i
7 4+94 + \sqrt{9} 40 ∑i=26(i+i)\sum_{i=2}^{6}(i+i) 297 ∑i=324i\sum_{i=3}^{24}i
8 49\sqrt{4}^{\sqrt{9}} 44 ∑i=29i\sum_{i=2}^{9}i 512 292^9
9 94\sqrt{9}^{\sqrt{4}} 60 ∑i=34(i!+i!)\sum_{i=3}^{4}(i!+i!) 528 ∑i=924(i+i)\sum_{i=9}^{24}(i+i)
10 4+(9)!4 + (\sqrt{9})! 64 434^3 570 ∑i=624(i+i)\sum_{i=6}^{24}(i+i)
11 4+9\sqrt{4} + 9 72 4!×94! \times \sqrt{9} 594 ∑i=324(i+i)\sum_{i=3}^{24}(i+i)
12 4×94 \times \sqrt{9} 77 ∑i=46i2\sum_{i=4}^{6}i^2 612 ∑i=34(i!×i!)\sum_{i=3}^{4}(i! \times i!)
13 4+94 + 9 78 ∑i=49(i+i)\sum_{i=4}^{9}(i+i) 714 ∑i=23((i!)!−i!)\sum_{i=2}^{3}((i!)!-i!)
14 ∑i=34(i+i)\sum_{i=3}^{4}(i+i) 81 94\sqrt{9^4} 717 ∑i=23((i!)!−i)\sum_{i=2}^{3}((i!)!-i)
15 4!−94! - 9 88 ∑i=29(i+i)\sum_{i=2}^{9}(i+i) 720 4×9!=6!\sqrt{4 \times 9}! = 6!
16 ∑i=23(i!+i!)\sum_{i=2}^{3}(i!+i!) 90 ∑i=26i2\sum_{i=2}^{6}i^2 722 ∑i=23(i!)!\sum_{i=2}^{3}(i!)!
18 4×9\sqrt{4} \times 9 114 ∑i=34(i×i!)\sum_{i=3}^{4}(i \times i!) 727 ∑i=23(i+(i!)!)\sum_{i=2}^{3}(i+(i!)!)
20 ∑i=26i\sum_{i=2}^{6}i 120 (9−4)!=5!(9-4)! = 5! 730 ∑i=23(i!+(i!)!)\sum_{i=2}^{3}(i!+(i!)!)
21 4!−94! - \sqrt{9} 121 ∑i=23(i!)!i!\sum_{i=2}^{3}\frac{(i!)!}{i!} 733 ∑i=23ii!\sum_{i=2}^{3}i^{i!}
22 ∑i=23(i×i!)\sum_{i=2}^{3}(i \times i!) 144 4!×(9)!4! \times (\sqrt{9})! 744 ∑i=23(i+i)!\sum_{i=2}^{3}(i+i)!
23 ∑i=34(i!−i)\sum_{i=3}^{4}(i!-i) 150 ∑i=46i!i\sum_{i=4}^{6}\frac{i!}{i} 1296 (9)!4(\sqrt{9})!^4
24 4×(9)!4 \times (\sqrt{9})! 153 ∑i=26i!i\sum_{i=2}^{6}\frac{i!}{i} 1444 ∑i=23((i!)!+(i!)!)\sum_{i=2}^{3}((i!)!+(i!)!)
25 ∑i=34i2\sum_{i=3}^{4}i^2 216 4!×94! \times 9 1728 ∑i=46(i!+i!)\sum_{i=4}^{6}(i!+i!)
27 4!+94! + \sqrt{9} 218 ∑i=23i!i!\sum_{i=2}^{3}\sqrt{i!}^{i!} 1744 ∑i=26(i!+i!)\sum_{i=2}^{6}(i!+i!)

2nd: 3 and 4 (77 values)

This pair benefits from 3!=63! = 6 and 4=2\sqrt{4} = 2, as well as 6!=7206! = 720.

Result Expression Result Expression Result Expression
0 ∑i=23(i−i)\sum_{i=2}^{3}(i-i) 31 ∑i=23ii\sum_{i=2}^{3}i^i 612 ∑i=34(i!×i!)\sum_{i=3}^{4}(i! \times i!)
1 4−34 - 3 36 (3!)4\sqrt{(3!)^4} 696 6!−4!6! - 4!
2 3!−43! - 4 37 ∑i=34(i+i!)\sum_{i=3}^{4}(i+i!) 714 ∑i=23((i!)!−i!)\sum_{i=2}^{3}((i!)!-i!)
3 34\sqrt{3}^{\sqrt{4}} 40 ∑i=26(i+i)\sum_{i=2}^{6}(i+i) 716 6!−46! - 4
4 3!−43! - \sqrt{4} 60 ∑i=34(i!+i!)\sum_{i=3}^{4}(i!+i!) 717 ∑i=23((i!)!−i)\sum_{i=2}^{3}((i!)!-i)
5 3+43 + \sqrt{4} 64 434^3 718 6!−46! - \sqrt{4}
6 3×43 \times \sqrt{4} 72 3×4!3 \times 4! 720 (3×4)!=6!(3 \times \sqrt{4})! = 6!
7 3+43 + 4 77 ∑i=46i2\sum_{i=4}^{6}i^2 722 6!+46! + \sqrt{4}
8 43\sqrt{4^3} 81 343^4 724 6!+46! + 4
9 34\sqrt{3^4} 90 ∑i=26i2\sum_{i=2}^{6}i^2 727 ∑i=23(i+(i!)!)\sum_{i=2}^{3}(i+(i!)!)
10 3!+43! + 4 114 ∑i=34(i×i!)\sum_{i=3}^{4}(i \times i!) 729 344!\sqrt[4]{3}^{4!}
12 3×43 \times 4 120 (3+4)!=5!(3+\sqrt{4})! = 5! 730 ∑i=23(i!+(i!)!)\sum_{i=2}^{3}(i!+(i!)!)
13 ∑i=23(i+i!)\sum_{i=2}^{3}(i+i!) 121 ∑i=23(i!)!i!\sum_{i=2}^{3}\frac{(i!)!}{i!} 733 ∑i=23ii!\sum_{i=2}^{3}i^{i!}
14 ∑i=34(i+i)\sum_{i=3}^{4}(i+i) 144 3!×4!3! \times 4! 744 6!+4!6! + 4!
15 ∑i=46i\sum_{i=4}^{6}i 150 ∑i=46i!i\sum_{i=4}^{6}\frac{i!}{i} 849 ∑i=46(i!−i)\sum_{i=4}^{6}(i!-i)
16 ∑i=23(i!+i!)\sum_{i=2}^{3}(i!+i!) 153 ∑i=26i!i\sum_{i=2}^{6}\frac{i!}{i} 852 ∑i=26(i!−i)\sum_{i=2}^{6}(i!-i)
18 4!−3!4! - 3! 180 6!4\frac{6!}{4} 864 ∑i=46i!\sum_{i=4}^{6}i!
20 ∑i=26i\sum_{i=2}^{6}i 218 ∑i=23i!i!\sum_{i=2}^{3}\sqrt{i!}^{i!} 872 ∑i=26i!\sum_{i=2}^{6}i!
21 4!−34! - 3 220 ∑i=23(i!)i\sum_{i=2}^{3}(i!)^i 879 ∑i=46(i+i!)\sum_{i=4}^{6}(i+i!)
22 ∑i=23(i×i!)\sum_{i=2}^{3}(i \times i!) 241 ∑i=23(i!)!i\sum_{i=2}^{3}\frac{(i!)!}{i} 892 ∑i=26(i+i!)\sum_{i=2}^{6}(i+i!)
23 ∑i=34(i!−i)\sum_{i=3}^{4}(i!-i) 283 ∑i=34ii\sum_{i=3}^{4}i^i 1296 (3!)4(3!)^4
24 3!×43! \times 4 285 ∑i=624i\sum_{i=6}^{24}i 1440 6!×46! \times \sqrt{4}
25 ∑i=34i2\sum_{i=3}^{4}i^2 297 ∑i=324i\sum_{i=3}^{24}i 1444 ∑i=23((i!)!+(i!)!)\sum_{i=2}^{3}((i!)!+(i!)!)
27 3+4!3 + 4! 360 6!4\frac{6!}{\sqrt{4}} 1728 ∑i=46(i!+i!)\sum_{i=4}^{6}(i!+i!)
29 ∑i=23ii!\sum_{i=2}^{3}\sqrt{i}^{i!} 570 ∑i=624(i+i)\sum_{i=6}^{24}(i+i) 1744 ∑i=26(i!+i!)\sum_{i=2}^{6}(i!+i!)
30 3!+4!3! + 4! 594 ∑i=324(i+i)\sum_{i=3}^{24}(i+i)

3rd: 2 and 9 (49 values)

This pair mainly uses 9=3\sqrt{9} = 3, which also gives us 3!=63! = 6.

Result Expression Result Expression Result Expression
0 ∑i=23(i−i)\sum_{i=2}^{3}(i-i) 22 ∑i=23(i×i!)\sum_{i=2}^{3}(i \times i!) 241 ∑i=23(i!)!i\sum_{i=2}^{3}\frac{(i!)!}{i}
1 9−2\sqrt{9} - 2 24 (3!−2)!=4!(3!-2)! = 4! 284 ∑i=29i2\sum_{i=2}^{9}i^2
2 3!−2\sqrt{3!-2} 29 ∑i=23ii!\sum_{i=2}^{3}\sqrt{i}^{i!} 512 292^9
3 92\sqrt[2]{9} 31 ∑i=23ii\sum_{i=2}^{3}i^i 714 ∑i=23((i!)!−i!)\sum_{i=2}^{3}((i!)!-i!)
4 3!−23! - 2 36 (3!)2(3!)^2 717 ∑i=23((i!)!−i)\sum_{i=2}^{3}((i!)!-i)
5 2+92 + \sqrt{9} 40 ∑i=26(i+i)\sum_{i=2}^{6}(i+i) 720 (2×9)!=6!(2 \times \sqrt{9})! = 6!
6 2×92 \times \sqrt{9} 44 ∑i=29i\sum_{i=2}^{9}i 722 ∑i=23(i!)!\sum_{i=2}^{3}(i!)!
7 9−29 - 2 64 262^6 727 ∑i=23(i+(i!)!)\sum_{i=2}^{3}(i+(i!)!)
8 232^3 81 929^2 730 ∑i=23(i!+(i!)!)\sum_{i=2}^{3}(i!+(i!)!)
9 92\sqrt{9^2} 88 ∑i=29(i+i)\sum_{i=2}^{9}(i+i) 733 ∑i=23ii!\sum_{i=2}^{3}i^{i!}
10 ∑i=23(i+i)\sum_{i=2}^{3}(i+i) 90 ∑i=26i2\sum_{i=2}^{6}i^2 744 ∑i=23(i+i)!\sum_{i=2}^{3}(i+i)!
11 2+92 + 9 120 (2+9)!=5!(2+\sqrt{9})! = 5! 852 ∑i=26(i!−i)\sum_{i=2}^{6}(i!-i)
12 2×3!2 \times 3! 121 ∑i=23(i!)!i!\sum_{i=2}^{3}\frac{(i!)!}{i!} 872 ∑i=26i!\sum_{i=2}^{6}i!
13 ∑i=23(i+i!)\sum_{i=2}^{3}(i+i!) 153 ∑i=26i!i\sum_{i=2}^{6}\frac{i!}{i} 892 ∑i=26(i+i!)\sum_{i=2}^{6}(i+i!)
16 ∑i=23(i!+i!)\sum_{i=2}^{3}(i!+i!) 218 ∑i=23i!i!\sum_{i=2}^{3}\sqrt{i!}^{i!} 1444 ∑i=23((i!)!+(i!)!)\sum_{i=2}^{3}((i!)!+(i!)!)
18 2×92 \times 9 220 ∑i=23(i!)i\sum_{i=2}^{3}(i!)^i 1744 ∑i=26(i!+i!)\sum_{i=2}^{6}(i!+i!)
20 ∑i=26i\sum_{i=2}^{6}i

4th: 4 and 5 (47 values)

Here, 5!=1205! = 120 and 4!=244! = 24 are particularly useful.

Result Expression Result Expression Result Expression
0 ∑i=25(i−i)\sum_{i=2}^{5}(i-i) 28 ∑i=25(i+i)\sum_{i=2}^{5}(i+i) 144 4!+5!4! + 5!
1 5−45 - 4 29 4!+54! + 5 152 ∑i=25i!\sum_{i=2}^{5}i!
2 ∑i=45ii\sum_{i=4}^{5}\frac{i}{i} 30 5!4\frac{5!}{4} 153 ∑i=45(i+i!)\sum_{i=4}^{5}(i+i!)
3 4+5\sqrt{4+5} 32 45\sqrt{4^5} 166 ∑i=25(i+i!)\sum_{i=2}^{5}(i+i!)
4 ∑i=25ii\sum_{i=2}^{5}\frac{i}{i} 33 ∑i=25i!i\sum_{i=2}^{5}\frac{i!}{i} 240 4×5!\sqrt{4} \times 5!
5 54\sqrt{5}^{\sqrt{4}} 41 ∑i=45i2\sum_{i=4}^{5}i^2 288 ∑i=45(i!+i!)\sum_{i=4}^{5}(i!+i!)
6 (5−4)!=3!(5-\sqrt{4})! = 3! 54 ∑i=25i2\sum_{i=2}^{5}i^2 290 ∑i=524i\sum_{i=5}^{24}i
7 4+5\sqrt{4} + 5 60 5!4\frac{5!}{\sqrt{4}} 304 ∑i=25(i!+i!)\sum_{i=2}^{5}(i!+i!)
9 4+54 + 5 96 5!−4!5! - 4! 480 4×5!4 \times 5!
10 4×5\sqrt{4} \times 5 116 5!−45! - 4 580 ∑i=524(i+i)\sum_{i=5}^{24}(i+i)
12 4!+5!\sqrt{4!+5!} 118 5!−45! - \sqrt{4} 625 545^4
14 ∑i=25i\sum_{i=2}^{5}i 120 4!×54! \times 5 696 ∑i=45(i×i!)\sum_{i=4}^{5}(i \times i!)
18 ∑i=45(i+i)\sum_{i=4}^{5}(i+i) 122 4+5!\sqrt{4} + 5! 718 ∑i=25(i×i!)\sum_{i=2}^{5}(i \times i!)
19 4!−54! - 5 124 4+5!4 + 5! 720 ((5−4)!)!=6!((5-\sqrt{4})!)! = 6!
20 4×54 \times 5 135 ∑i=45(i!−i)\sum_{i=4}^{5}(i!-i) 1024 454^5
25 54\sqrt{5^4} 138 ∑i=25(i!−i)\sum_{i=2}^{5}(i!-i)

5th: 0 and 9 (45 values)

This pair combines 0!=10! = 1 with the versatility of 9=3\sqrt{9} = 3.

Result Expression Result Expression Result Expression
0 0×90 \times 9 21 ∑i=16i\sum_{i=1}^{6}i 242 ∑i=13(i!)!i\sum_{i=1}^{3}\frac{(i!)!}{i}
1 909^0 23 ∑i=13(i×i!)\sum_{i=1}^{3}(i \times i!) 285 ∑i=19i2\sum_{i=1}^{9}i^2
2 9−0!\sqrt{9} - 0! 24 (0!+9)!=4!(0!+\sqrt{9})! = 4! 714 ∑i=13((i!)!−i!)\sum_{i=1}^{3}((i!)!-i!)
3 0+9\sqrt{0+9} 30 ∑i=13ii!\sum_{i=1}^{3}\sqrt{i}^{i!} 717 ∑i=13((i!)!−i)\sum_{i=1}^{3}((i!)!-i)
4 0!+90! + \sqrt{9} 32 ∑i=13ii\sum_{i=1}^{3}i^i 720 (0+3!)!=6!(0+3!)! = 6!
5 3!−0!3! - 0! 41 ∑i=13(i!×i!)\sum_{i=1}^{3}(i! \times i!) 723 ∑i=13(i!)!\sum_{i=1}^{3}(i!)!
6 (0+9)!=3!(0+\sqrt{9})! = 3! 42 ∑i=16(i+i)\sum_{i=1}^{6}(i+i) 729 ∑i=13(i+(i!)!)\sum_{i=1}^{3}(i+(i!)!)
7 0!+3!0! + 3! 45 ∑i=19i\sum_{i=1}^{9}i 732 ∑i=13(i!+(i!)!)\sum_{i=1}^{3}(i!+(i!)!)
8 9−0!9 - 0! 90 ∑i=19(i+i)\sum_{i=1}^{9}(i+i) 734 ∑i=13ii!\sum_{i=1}^{3}i^{i!}
9 0+90 + 9 91 ∑i=16i2\sum_{i=1}^{6}i^2 746 ∑i=13(i+i)!\sum_{i=1}^{3}(i+i)!
10 0!+90! + 9 120 (3!−0!)!=5!(3!-0!)! = 5! 852 ∑i=16(i!−i)\sum_{i=1}^{6}(i!-i)
12 ∑i=13(i+i)\sum_{i=1}^{3}(i+i) 122 ∑i=13(i!)!i!\sum_{i=1}^{3}\frac{(i!)!}{i!} 873 ∑i=16i!\sum_{i=1}^{6}i!
14 ∑i=13i2\sum_{i=1}^{3}i^2 154 ∑i=16i!i\sum_{i=1}^{6}\frac{i!}{i} 894 ∑i=16(i+i!)\sum_{i=1}^{6}(i+i!)
15 ∑i=13(i+i!)\sum_{i=1}^{3}(i+i!) 219 ∑i=13i!i!\sum_{i=1}^{3}\sqrt{i!}^{i!} 1446 ∑i=13((i!)!+(i!)!)\sum_{i=1}^{3}((i!)!+(i!)!)
18 ∑i=13(i!+i!)\sum_{i=1}^{3}(i!+i!) 221 ∑i=13(i!)i\sum_{i=1}^{3}(i!)^i 1746 ∑i=16(i!+i!)\sum_{i=1}^{6}(i!+i!)

Why are some pairs more useful?

Particularly useful digits

Digit Useful properties Available values
4 4=2\sqrt{4} = 2 (integer), 4!=244! = 24 2, 4, 24
9 9=3→3!=6→6!=720\sqrt{9} = 3 \to 3! = 6 \to 6! = 720 3, 6, 9, 720
3 3!=6→6!=7203! = 6 \to 6! = 720 3, 6, 720
5 5!=1205! = 120 5, 120
0 0!=10! = 1 0, 1

Less versatile digits

Digit Limitation
1 1!=11! = 1, 1=1\sqrt{1} = 1 (unchanged)
2 2\sqrt{2} is not an integer
7 7\sqrt{7} is not an integer, 7!=50407! = 5040 (too large)
8 8\sqrt{8} is not an integer

A rule of thumb

When choosing a pair of digits, choose one that includes 4 or 9. Their square roots, 4=2\sqrt{4}=2 and 9=3\sqrt{9}=3, can be used in factorials to create further values.

Frequently asked questions

Why are 4 and 9 the best pair?

Both have whole-number square roots:

  • 4=2\sqrt{4} = 2 → use it as an exponent or a summation bound.
  • 9=3\sqrt{9} = 3 → apply factorials to obtain 3!=63! = 6 and 6!=7206! = 720.

Having 2 and 3 available lets you construct every value from 1 to 10 and build larger values as well.

What is sigma notation?

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

For example:

  • ∑i=14i=1+2+3+4=10\sum_{i=1}^{4}i = 1 + 2 + 3 + 4 = 10
  • ∑i=23i2=22+32=4+9=13\sum_{i=2}^{3}i^2 = 2^2 + 3^2 = 4 + 9 = 13
  • ∑i=23i!=2!+3!=2+6=8\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 !!, is the product of the integers from 1 through the given number.

For example:

  • 3!=1×2×3=63! = 1 \times 2 \times 3 = 6
  • 4!=1×2×3×4=244! = 1 \times 2 \times 3 \times 4 = 24
  • 5!=1×2×3×4×5=1205! = 1 \times 2 \times 3 \times 4 \times 5 = 120
  • 6!=7206! = 720

A special case is 0!=10! = 1, by mathematical definition.

Learn more in What is a factorial?.

Which pairs are the least useful?

The pairs that produce the fewest values are:

  • (0, 1) and (1, 1), which produce only three values.

How can I use this information?

When you need to build a result from two digits:

  1. Look for 4 or 9 and try 4=2\sqrt{4}=2 or 9=3\sqrt{9}=3.
  2. Look for 3 or 5 and try factorials: 3!=63!=6 or 5!=1205!=120.
  3. If the target is near 720, look for a way to make 6 and then use 6!6!.

Don’t forget to follow us on Facebook!

Share:
  • Share on Facebook
  • Share on X

More articles

All articles
enth