# Make numbers 1-100 with only four 8s

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:

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

• Do the exponents have to be one of the four eights or can they be any number? Commented Oct 4, 2017 at 0:20
• And do you require exactly four 8s or just no more than 4 8s. Commented Oct 4, 2017 at 1:42
• And, are you allowed brackets? Commented Oct 4, 2017 at 1:43

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).

• Filling your first two gaps: 18 = 8 + 8 + $\lceil\sqrt{\sqrt{\sqrt{88}}}\rceil$, 20 = 8 + 8 + $\lceil\sqrt{\sqrt{88}}\rceil$.
– h34
Commented Oct 4, 2017 at 19:21
• Filling your last gap: 100 = $\lfloor\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{8^8}}}}}}} * 88\rfloor$.
– h34
Commented Oct 4, 2017 at 19:30
• And now there's only one gap left to fill, 76, for which an answer is $\lceil\sqrt(8*8*88)\rceil$.
– h34
Commented Oct 5, 2017 at 12:36
• What about 100 = 88/.88 ? Commented Mar 20, 2022 at 9:26

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.

• 8 = ( 8 ^ (8 - 8) ) * 8
– Joe
Commented Oct 4, 2017 at 2:33
• Also: 6 = 8 - ( 8 + 8 ) / 8 ; 10 = 8 + ( 8 + 8 ) / 8
– Joe
Commented Oct 4, 2017 at 2:40
• 10 = (88 - 8)/8, if "concatenation" counts Commented Oct 4, 2017 at 2:53
• Are we allowed decimals? $\frac{8}{.8} + \frac{8}{8} = 11$ Commented Oct 4, 2017 at 13:55

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!

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 : ?