Work in progress. 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 = 40 = $8 \times \sqrt{8+8} + 8$ 41 = 42 = 43 = 44 = $\frac{88}{\sqrt{\sqrt{8+8}}}$ 45 = 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 = 50 = 51 = 52 = $\frac{88}{\lfloor\sqrt8\rfloor} + 8$ 53 = $8 \times 8 - 8 - \lceil \sqrt8 \rceil$ 54 = $8 \times 8 - 8 - \lfloor \sqrt8 \rfloor$ 55 = **56** = $8 \times (8 - \frac88)$ 57 = 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 = 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 =