# 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? Oct 4 '17 at 0:20
• And do you require exactly four 8s or just no more than 4 8s. Oct 4 '17 at 1:42
• And, are you allowed brackets? Oct 4 '17 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
Oct 4 '17 at 19:21
• Filling your last gap: 100 = $\lfloor\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{8^8}}}}}}} * 88\rfloor$.
– h34
Oct 4 '17 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
Oct 5 '17 at 12:36
• @h34. Thanks. The list is now complete. Oct 5 '17 at 12:49

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
Oct 4 '17 at 2:33
• Also: 6 = 8 - ( 8 + 8 ) / 8 ; 10 = 8 + ( 8 + 8 ) / 8
– Joe
Oct 4 '17 at 2:40
• 10 = (88 - 8)/8, if "concatenation" counts Oct 4 '17 at 2:53
• Are we allowed decimals? $\frac{8}{.8} + \frac{8}{8} = 11$ Oct 4 '17 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!

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  : ?