Htydjtk

Create all numbers 1 - 100 using equations made up of 1,9,6,8.

Rules:

• Use all four digits exactly once


• Allowed operations: +, -, x, ÷, ! (factorial), exponentiation, square root.


• Parentheses and grouping (e.g. "21") are also allowed.


• You have to keep the order 1,9,6,8 for all numbers.


• Exponentiation can only be used in the number order with the numbers provided. Eg. 1^9 + 6 + 8 is allowed. Not 1^6 + 9 + 8.


• The modulus operator is not allowed.


• Rounding is not allowed (e.g. 201/8=25).


• Decimal point is allowed.

Credit to Fitch496 for the idea.

Partial answer (the most obvious solutions):

1 = -1^9 - 6 + 8 = 1^968

2 = 1 + sqrt(9) + 6 - 8

5 = 19 - 6 - 8

6 = -1 + 9 + 6 - 8 = 1 - 9 + 6 + 8

7 = 1 * (9 + 6) - 8

8 = 1 + 9 + 6 - 8

9 = 1^96 + 8

11 = -1 + 96 / 8

12 = 1 * 96 / 8 = 1 + 9 - 6 + 8

13 = 1 + 96 / 8

16 = (1 - 9) * (6 - 8)

17 = 19 + 6 - 8

18 = 1 + sqrt(9) + 6 + 8

21 = 19 - 6 + 8

22 = -1 + 9 + 6 + 8

23 = 1 * (9 + 6) + 8

24 = 1 + 9 + 6 + 8

25 = -1 + sqrt(9) * 6 + 8

26 = 1 * sqrt(9) * 6 + 8

27 = 1 + sqrt(9) * 6 + 8

29 = -19 + 6 * 8

31 = 19 + 6 + 8

32 = (1 + sqrt(9)) * 6 + 8

52 = (1 + 9) * 6 - 8

57 = 1 * 9 + 6 * 8

58 = -1 - 9 + 68

59 = -1 * 9 + 68

60 = 1 - 9 + 68

62 = 1 * 9 * 6 + 8

63 = 1 + 9 * 6 + 8

67 = 19 + 6 * 8

68 = (1 + 9) * 6 + 8

76 = -1 + 9 + 68

77 = 1 * 9 + 68

78 = 1 + 9 + 68

80 = -1 - 9 + 6! / 8

81 = -1 * 9 + 6! / 8

82 = 1 - 9 + 6! / 8

87 = -1 + 96 - 8

88 = 1 * 96 - 8

89 = 1 + 96 - 8

98 = -1 + 9 + 6! / 8

99 = 1 * 9 + 6! / 8

100 = 1 + 9 + 6! / 8

Partial Answer (too lazy to type them out :P)

Here are all the numbers which have not been obtained already by trolley813, Omega Krypton and ppgdev

$28 = 1 times ((sqrt{9})! times 6) - 8$
$30 =-(1times(sqrt{9})!) + sqrt{sqrt{6^8}} $
$33 = 19 + 6 + 8$
$34 = ((1+(sqrt{9})!) times 6) - 8 $
$35 = (1 times sqrt{(sqrt{9})^6}) + 8$
$36 = (1 + sqrt{(sqrt{9})^6}) + 8$
$37 = 1^9 + sqrt{sqrt{6^8}}$
$38 = (1+sqrt{9})! + 6 + 8$
$41 = -(1 + (sqrt{9})!) + (6 times 8)$
$42 = -(1 times (sqrt{9})!) + (6 times 8)$
$43 = -1 + ((sqrt{9})! times 6) + 8$
$44 = 1 times ((sqrt{9})! times 6) + 8$
$45 = 1 + ((sqrt{9})! times 6) + 8$
$ 55 = 1 + (sqrt{9})! + (6 times 8)$
$56 = (1^9 + 6) times 8 $
$ 64 = ((-1 + sqrt{9}) + 6) times 8$
$65 = -(1 times sqrt{9}) + 68 $
$66 = 1 - sqrt{9} + 68 $
$69 = 1^9 + 68$
$70 = -1 + sqrt{9} + 68 $
$71 = -1 + ((sqrt{9} + 6)times 8)$
$72 = ((1 times sqrt{9}) + 6) times 8$
$73 = 1 + ((sqrt{9} + 6)times 8) $
$74 = (1 times (sqrt{9})!) + 68$
$75 = 1 + (sqrt{9})!) + 68$
$ 79 = -1 + ((9 + sqrt{sqrt{ldots sqrt{6}}}) times 8)$
$ 86 = -1 - sqrt{9} + (6!/8)$
$90 = (1^9 times 6!)/8 $
$91 = 1^9 + (6!/8)$

Here is an answer for 39 that was missing so far:

39 = -(1*9) + (6*8)

I also got 79

79 = ((-1 + (sqrt(9)!))!! + (8!!)/6