5
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Make numbers 1, 2, 3, 4, 5,.....100 using only four 8s. You can use multiplication, division, addition, and subtraction. No negatives. You may use exponents.

Example:

$1 = 8 \times 8 \div 8 \div 8$

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  • $\begingroup$ Do the exponents have to be one of the four eights or can they be any number? $\endgroup$ – Travis Oct 4 '17 at 0:20
  • 1
    $\begingroup$ And do you require exactly four 8s or just no more than 4 8s. $\endgroup$ – Penguino Oct 4 '17 at 1:42
  • $\begingroup$ And, are you allowed brackets? $\endgroup$ – Penguino Oct 4 '17 at 1:43
7
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It is not possible to generate all the numbers with only the give operations.
I used a computer for that.
The only ones are 1,2,3,4,6,7,8,9,10,12,15,16,17,19,24,32,48,56,63,64,65,72,80,87,88,89 (marked in bold).

So I took the liberty to use other functions like square root ($\sqrt{x}$), ceil ($\lceil x \rceil$), floor ($\lfloor x \rfloor$).

1 = $\frac{88}{88}$
2 = $\frac88 + \frac88$
3 = $\frac{88}{8} - 8$
4 = $\frac{8 \times 8}{8+8}$
5 = $\sqrt{8+8} + \frac88$
6 = $8 - \frac{8 + 8}{8}$
7 = $\frac{8 \times 8 - 8}{8}$
8 = $8 + 8 \times (8 - 8)$
9 = $\frac{8 \times 8 + 8}{8}$
10 = $\frac{88 - 8}{8}$
11 = $\lceil \sqrt8 \rceil + 8-8+8$
12 = $\frac{88 + 8}{8}$
13 = $\lfloor \sqrt8 \rfloor + \frac{88}{8}$
14 = $\lceil \sqrt8 \rceil + \frac{88}{8}$
15 = $8 + 8 - \frac88$
16 = $8 \times \frac{8+8}{8}$
17 = $8 + 8 + \frac88$
18 = $8 + 8 + \sqrt{\sqrt{8+8}}$
19 = $\frac{88}{8} + 8$
20 = $8 + 8 + \sqrt{8+8}$
21 = $8 + 8 + \lceil \sqrt{8} \rceil + \lfloor \sqrt{8} \rfloor$
22 = $\frac{88}{\sqrt{8+8}}$
23 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} - \sqrt{8+8}$
24 = $88-8\times 8$
25 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} - \sqrt{\sqrt{8+8}}$
26 = $ 8 +8 +8 + \lfloor \sqrt{8} \rfloor$
27 = $\lceil \sqrt8 \rceil + 8+8+8$
28 = $8 \times \lceil\sqrt8\rceil + \sqrt{8+8}$
29 = $8 \times \sqrt{8+8} - \lceil \sqrt8 \rceil$
30 = $8 \times \sqrt{8+8} - \lfloor \sqrt8 \rfloor$
31 = $\lceil\sqrt{8}\rceil^{\lceil\sqrt{8}\rceil} + \sqrt{8+8}$
32 = $8 + 8 + 8 + 8$
33 = $8 \times \sqrt{8+8} + \lfloor \sqrt{\sqrt8} \rfloor$
34 = $8 \times \sqrt{8+8} + \lfloor \sqrt8 \rfloor$
35 = $8 \times \sqrt{8+8} + \lceil \sqrt8 \rceil$
36 = $(8 + \lfloor\sqrt{\sqrt8}\rfloor) \times \sqrt{8+8}$
37 = $\lfloor\frac{88}{\lceil\sqrt8\rceil}\rfloor + 8$
38 = $\lceil\frac{88}{\lceil\sqrt8\rceil}\rceil + 8$
39 = $(8+ \lceil\sqrt8\rceil + \lfloor\sqrt8\rfloor) \times \lceil\sqrt8\rceil$
40 = $8 \times \sqrt{8+8} + 8$
41 = $\frac{88}{\lfloor\sqrt8\rfloor} - \lceil\sqrt8\rceil$
42 = $\frac{88}{\lfloor\sqrt8\rfloor} - \lfloor\sqrt8\rfloor$
43 = $\frac{88}{\lfloor\sqrt8\rfloor} - \lfloor\sqrt{\sqrt8}\rfloor$
44 = $\frac{88}{\sqrt{\sqrt{8+8}}}$
45 = $\frac{88}{\lfloor\sqrt8\rfloor} + \lfloor\sqrt{\sqrt8}\rfloor$
46 = $\frac{88}{\lfloor\sqrt8\rfloor} + \lfloor\sqrt8\rfloor$
47 = $\frac{88}{\lfloor\sqrt8\rfloor} + \lceil\sqrt8\rceil$
48 = $8 \times 8 - 8 -8$
49 = $(8 - \lfloor \sqrt{\sqrt8} \rfloor) \times (8 - \lfloor \sqrt{\sqrt8} \rfloor)$
50 = $(\lfloor \sqrt8 \rfloor + \lceil \sqrt8 \rceil) \times (8 + \lfloor \sqrt8 \rfloor)$
51 = $(8+8) \times \lceil \sqrt8 \rceil + \lceil \sqrt8 \rceil$
52 = $\frac{88}{\lfloor\sqrt8\rfloor} + 8$
53 = $8 \times 8 - 8 - \lceil \sqrt8 \rceil$
54 = $8 \times 8 - 8 - \lfloor \sqrt8 \rfloor$
55 = $8 \times 8 - 8 - \lfloor \sqrt{\sqrt8} \rfloor$
56 = $8 \times (8 - \frac88)$
57 = $8 \times 8 - 8 + \lfloor \sqrt{\sqrt8} \rfloor$
58 = $8 \times 8 - 8 + \lfloor \sqrt8 \rfloor$
59 = $8 \times 8 - 8 + \lceil \sqrt8 \rceil$
60 = $8 \times 8 - \sqrt{8+8}$
61 = $8\times 8 - \lceil \frac{8}{\sqrt8}\rceil$
62 = $8\times 8 - \sqrt{\sqrt{8+8}}$
63 = $8 \times 8 - \frac88$
64 = $8 \times 8 \times \frac88$
65 = $8 * 8 + \frac88$
66 = $8\times 8 + \sqrt{\sqrt{8+8}}$
67 = $8\times 8 + \lceil \frac{8}{\sqrt8}\rceil$
68 = $8 \times 8 + \sqrt{8+8}$
69 = $8 + 8 \times 8 - \lceil \sqrt8 \rceil$
70 = $8 \times 8 + \sqrt{8+8}$
71 = $8 +8 \times 8 - \lfloor \sqrt{\sqrt8} \rfloor$
72 = $88 - 8 - 8$
73 = $\lfloor \sqrt{\sqrt8} \rfloor + 8 +8 \times 8$
74 = $\lfloor \sqrt8 \rfloor + 8 +8 \times 8$
75 = $\lceil \sqrt8 \rceil + 8 +8 \times 8$
76 = $\lceil \sqrt{88 \times 8 \times 8} \rceil$ - thanks to h34
77 = $88 - 8 - \lceil\sqrt8\rceil$
78 = $88 - 8 - \lfloor\sqrt8\rfloor$
79 = $\lceil\sqrt8\rceil^{\sqrt{8+8}} - \lfloor\sqrt8\rfloor$
80 = $8\times 8 + 8 + 8$
81 = $(\lfloor\sqrt8\rfloor + \lfloor\sqrt{\sqrt8}\rfloor)^{\sqrt{8+8}}$
82 = $88 - 8 + \lfloor\sqrt8\rfloor$
83 = $88 - 8 + \lceil\sqrt8\rceil$
84 = $88 - \sqrt{8+8}$
85 = $88 - \lceil\frac{8}{\sqrt8}\rceil$
86 = $88 - \lfloor\frac{8}{\sqrt8}\rfloor$
87 = $88 - \frac88$
88 = $88 - 8 + 8$
89 = $88 + \frac88$
90 = $88 + \lfloor\frac{8}{\sqrt8}\rfloor$
91 = $88 + \lceil\frac{8}{\sqrt8}\rceil$
92 = $88 + \sqrt{8+8}$
93 = $88 +8 - \lceil\sqrt8\rceil$
94 = $88 +8 - \lfloor\sqrt8\rfloor$
95 = $88 +8 - \lfloor\sqrt{\sqrt8}\rfloor$
96 = $8 \times \sqrt{8+8} \times \lceil \sqrt8 \rceil$
97 = $88 +8 + \lfloor\sqrt{\sqrt8}\rfloor$
98 = $88 +8 + \lfloor\sqrt8\rfloor$
99 = $88 +8 + \lceil\sqrt8\rceil$
100 = $(8 + \lfloor \sqrt8 \rfloor)^{\sqrt{\sqrt{8+8}}}$

Bonus:
If we allow log functions we can generate every number like this:

$x = \log_{\frac{\lfloor\sqrt{\sqrt8}\rfloor}{\lfloor\sqrt8\rfloor}}\left({\log_8\underbrace{\sqrt{\sqrt{\dots\sqrt{8\,}\,}\,}}_\text{x square roots}}\right)$

This is equivalent to

$x = \log_{\frac12}\left({\log_8{8^{\frac{1}{2^x}}}}\right)$

Going further:

$x = \log_{\frac12}\left({\frac{1}{2^x}}\right)$

Depending on he number of square roots, we can get any number.

And the snarky solution:

Turn one eight 90 degrees to get: $\infty$ and then $\frac{8-8}{8} \times \infty=0 \times \infty$ which can be equal to anything. (l know, I know, you will say it's undefined, but think... limits).

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  • $\begingroup$ Filling your first two gaps: 18 = 8 + 8 + $\lceil\sqrt{\sqrt{\sqrt{88}}}\rceil$, 20 = 8 + 8 + $\lceil\sqrt{\sqrt{88}}\rceil$. $\endgroup$ – h34 Oct 4 '17 at 19:21
  • $\begingroup$ Filling your last gap: 100 = $\lfloor\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{8^8}}}}}}} * 88\rfloor$. $\endgroup$ – h34 Oct 4 '17 at 19:30
  • 2
    $\begingroup$ And now there's only one gap left to fill, 76, for which an answer is $\lceil\sqrt(8*8*88)\rceil$. $\endgroup$ – h34 Oct 5 '17 at 12:36
  • $\begingroup$ @h34. Thanks. The list is now complete. $\endgroup$ – Marius Oct 5 '17 at 12:49
3
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I'm not sure if this is doable with only additions, subtractions, multiplications, divisions and exponents. This is what I managed to get:

1  = (8 * 8) / (8 * 8) 
2  = (8 / 8) + (8 / 8)
3  = (8 + 8 + 8) / 8
4  = (8 * 8) / (8 + 8)
5  = ?
6  = 8 - (8 + 8) / 8
7  = (8 * 8 - 8) / 8
8  = (8 ^ (8 - 8)) * 8
9  = (8 * 8 + 8) / 8
10 = 8 + (8 + 8) / 8
11 = ?
12 = (88 + 8) / 8
13 = ?
14 = ?
15 = 8 + 8 - 8 / 8
16 = 8 + 8 * 8 / 8
17 = 8 + 8 + 8 / 8
18 = ?
19 = 88 / 8 + 8
20 = ?
21 = ?
22 = ?
23 = ?
24 = 88 - 8 * 8
25 = ?
26 = ?
27 = ?
28 = ?
29 = ?
30 = ?
31 = ?
32 = 8 + 8 + 8 + 8
33 = ?
34 = ?
35 = ?
36 = ?
37 = ?
38 = ?
39 = ?
40 = ?
41 = ?
42 = ?
43 = ?
44 = ?
45 = ?
46 = ?
47 = ?
48 =  8 * 8 - 8 - 8
49 = ?
50 = ?
51 = ?
52 = ?
53 = ?
54 = ?
55 = ?
56 = (8 - 8 / 8) * 8
57 = ?
58 = ?
59 = ?
60 = ?
61 = ?
62 = ?
63 = 8 * 8 - 8 / 8
64 = 8 * 8 * 8 / 8
65 = 8 * 8 + 8 / 8
66 = ?
67 = ?
68 = ?
69 = ?
70 = ?
71 = ?
72 = (8 + 8 / 8) * 8
73 = ?
74 = ? 
75 = ?
76 = ?
77 = ?
78 = ?
79 = ?
80 = 8 * 8 + 8 + 8
81 = ?
82 = ?
83 = ?
84 = ?
85 = ?
86 = ?
87 = 88 - 8 / 8
88 = 88 * 8 / 8
89 = 88 + 8 / 8
90 = ?
91 = ?
92 = ?
93 = ?
94 = ?
95 = ?
96 = ?
97 = ?
98 = ?
99 = ?
100 = ?

(Thank you to Joe for the help with 6, 8 and 10, and kushj for the some of the 11-100 entries)

I do have some questions regarding the rules, though - because if we're allowed some fun takes on the 'exponents' rule, these kind of approaches become feasible:

6  = 8 - Log8(8 * 8)
10 = 8 + Log8(8 * 8)

Having the sqrt operator would be very useful too, but I don't know whether it's allowed.

5  = Sqrt(8 + 8) + (8 / 8)
8  = Sqrt(8 * 8) * (8 / 8)

Please do tell me whether these two operators are allowed for the puzzle.

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  • $\begingroup$ 8 = ( 8 ^ (8 - 8) ) * 8 $\endgroup$ – Joe Oct 4 '17 at 2:33
  • $\begingroup$ Also: 6 = 8 - ( 8 + 8 ) / 8 ; 10 = 8 + ( 8 + 8 ) / 8 $\endgroup$ – Joe Oct 4 '17 at 2:40
  • $\begingroup$ 10 = (88 - 8)/8, if "concatenation" counts $\endgroup$ – kushj Oct 4 '17 at 2:53
  • $\begingroup$ Are we allowed decimals? $\frac{8}{.8} + \frac{8}{8} = 11$ $\endgroup$ – Trenin Oct 4 '17 at 13:55
2
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Wrote a script to generate quick answer using only brackets and 4 basic operators.

Now on to much more interesting task to generate rest of numbers!

I can't seem to put it in spoiler tags at once, so Mod help would be appreciated.

1  :  (8 + 8 - 8) / 8
2  :  8 / 8 + 8 / 8
3  :  (8 + 8 + 8) / 8
4  :  8 / (8 + 8) * 8
5  : ?
6  :  8 - (8 + 8) / 8
7  :  (8 * 8 - 8) / 8
8  :  8 + (8 - 8) * 8
9  :  (8 + 8 * 8) / 8
10  :  8 + (8 + 8) / 8
11  : ?
12  : ?
13  : ?
14  : ?
15  :  8 + 8 - 8 / 8
16  :  8 + 8 + 8 - 8
17  :  8 + 8 + 8 / 8
18  : ?
19  : ?
20  : ?
21  : ?
22  : ?
23  : ?
24  : ?
25  : ?
26  : ?
27  : ?
28  : ?
29  : ?
30  : ?
31  : ?
32  :  8 + 8 + 8 + 8
33  : ?
34  : ?
35  : ?
36  : ?
37  : ?
38  : ?
39  : ?
40  : ?
41  : ?
42  : ?
43  : ?
44  : ?
45  : ?
46  : ?
47  : ?
48  :  8 * 8 - 8 - 8
49  : ?
50  : ?
51  : ?
52  : ?
53  : ?
54  : ?
55  : ?
56  :  (8 - 8 / 8) * 8
57  : ?
58  : ?
59  : ?
60  : ?
61  : ?
62  : ?
63  :  8 * 8 - 8 / 8
64  :  (8 + 8 - 8) * 8
65  :  8 * 8 + 8 / 8
66  : ?
67  : ?
68  : ?
69  : ?
70  : ?
71  : ?
72  :  (8 + 8 / 8) * 8
73  : ?
74  : ?
75  : ?
76  : ?
77  : ?
78  : ?
79  : ?
80  :  8 + 8 + 8 * 8
81  : ?
82  : ?
83  : ?
84  : ?
85  : ?
86  : ?
87  : ?
88  : ?
89  : ?
90  : ?
91  : ?
92  : ?
93  : ?
94  : ?
95  : ?
96  : ?
97  : ?
98  : ?
99  : ?
100  : ?
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