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:
$1 = 8 \times 8 \div 8 \div 8$
Answer
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).
No comments:
Post a Comment