Calculation - the hard way

Question

This is a puzzle I love to play with my math students and I hope you will enjoy it too:

You are given the numbers 1, 2, 3, 4, and 5 exactly once.

Your target is a number, e.g. 36. Can you create a calculation with addition, subtraction, multiplication, division and parentheses, so that you arrive at this number?

Concatenating the numbers (like 12 out of 1 and 2) is explicitly forbidden.

For this example, the solution would be

$$ 36=(2+4) \cdot (1+5) $$

but remember: You can use each number only once.

Can you find a calculation for all numbers from 1 to 75?

BONUS question: Now powers are allowed! Can you go to 125 now?

Answer 1

1 = 1  
2 = 2  
3 = 3  
4 = 4  
5 = 5  
6 = 5 + 1  
7 = 5 + 2  
8 = 5 + 3  
9 = 5 + 4  
10 = 5 + 4 + 1  
11 = 5 + 4 + 2  
12 = 5 + 4 + 3  
13 = 5 + 4 + 3 + 1  
14 = 5 + 4 + 3 + 2  
15 = 5 + 4 + 3 + 2 + 1  
16 = 4 ⋅ (1 + 3)  
17 = (5 ⋅ 4) - 3  
18 = (5 ⋅ 4) - 2  
19 = (5 ⋅ 4) - 1  
20 = (5 ⋅ 4)  
21 = (5 ⋅ 4) + 1  
22 = (5 ⋅ 4) + 2  
23 = (5 ⋅ 4) + 3  
24 = (5 + 1) ⋅ 4  
25 = ((4 + 5) ⋅ 3) - 2  
26 = ((4 + 5) ⋅ 3) - 1  
27 = ((4 + 5) ⋅ 3)  
28 = ((4 + 5) ⋅ 3) + 1  
29 = ((4 + 5) ⋅ 3) + 2  
30 = 2 ⋅ 3 ⋅ 5  
31 = (2 ⋅ 3 ⋅ 5) + 1  
32 = (1 + 3) ⋅ 4 ⋅ 2  
33 = (2 ⋅ 3 ⋅ 5) + 4 - 1  
34 = (2 ⋅ 3 ⋅ 5) + 4  
35 = (2 ⋅ 3 ⋅ 5) + 4 + 1  
36 = (2 + 4) ⋅ (5 + 1)
37 = ((4 + 3) ⋅ 5) + 2  
38 = ((4 + 3) ⋅ 5) + 2 + 1  
40 = 2 ⋅ 4 ⋅ 5  
41 = (2 ⋅ 4 ⋅ 5) + 1  
42 = (2 ⋅ 4 ⋅ 5) - 1 + 3  
43 = (2 ⋅ 4 ⋅ 5) + 3  
44 = (2 ⋅ 4 ⋅ 5) + 3 + 1  
45 = (2 + 3 + 4) ⋅ 5  
46 = ((2 + 3 + 4) ⋅ 5) + 1  
47 = ((2 + 4) ⋅ (3 + 5)) - 1  
48 = ((2 + 4) ⋅ (3 + 5))  
49 = ((2 + 4) ⋅ (3 + 5)) + 1  
50 = 2 ⋅ 5 ⋅ (4 + 1)  
51 = ((5 ⋅ 4) - 2 - 1) ⋅ 3  
52 = ((5 ⋅ 2) + 3) ⋅ 4  
53 = (((5 ⋅ 2) + 3) ⋅ 4) + 1  
54 = (5 + 4) ⋅ 3 ⋅ 2  
55 = ((5 + 4) ⋅ 3 ⋅ 2) + 1  
56 = (4 + 2 + 1) ⋅ (5 + 3)  
57 = (5 ⋅ 4 ⋅ 3) - 2 - 1  
58 = (5 ⋅ 4 ⋅ 3) - 2  
59 = (5 ⋅ 4 ⋅ 3) - 1  
60 = (5 ⋅ 4 ⋅ 3)  
61 = (5 ⋅ 4 ⋅ 3) + 1  
62 = (5 ⋅ 4 ⋅ 3) + 2  
63 = (5 ⋅ 4 ⋅ 3) + 1 + 2  
64 = (5 + 3) ⋅ 4 ⋅ 2  
65 = ((5 + 3) ⋅ 4 ⋅ 2) + 1  
66 = (5 + 1) ⋅ ((4 ⋅ 2) + 3)  
67 = ((5 ⋅ 3) + 2) ⋅ 4) - 1  
68 = ((5 ⋅ 3) + 2) ⋅ 4)  
69 = ((5 ⋅ 3) + 2) ⋅ 4) + 1  
70 = (3 + 4) ⋅ 5 ⋅ 2  
71 = ((3 + 4) ⋅ 5 ⋅ 2) + 1  
72 = (5 + 1) ⋅ 3 ⋅ 4  
73 = ((4 + 1) ⋅ 3 ⋅ 5) - 2  
74 = ((5 + 1) ⋅ 3 ⋅ 4) + 2  
75 = (4 + 1) ⋅ 3 ⋅ 5

These are all the solutions for the main puzzle, I'm going to give the bonus puzzle a go when I wake up.

Answer 2

With some dynamic programming we can find all possibilities. Without exponents, we can go to exactly 75:

1: 1 * (2 / (3 + (4 - 5)))
2: (((4 + 5) - 3) / 2) - 1
3: 1 * (((4 + 5) - 3) / 2)
4: (2 + (3 * (5 - 4))) - 1
5: 1 * (2 + (3 * (5 - 4)))
6: 1 + (2 + (3 * (5 - 4)))
7: 1 * (2 + (5 * (4 - 3)))
8: 1 + (2 + (5 * (4 - 3)))
9: 1 + (2 * (3 - (4 - 5)))
10: 1 * (2 * (5 * (4 - 3)))
11: 1 + (2 * (5 * (4 - 3)))
12: 1 * (2 * ((4 + 5) - 3))
13: 1 + (2 * ((4 + 5) - 3))
14: 1 * (2 * ((3 * 4) - 5))
15: 1 + (2 * ((3 * 4) - 5))
16: ((4 + (3 * 5)) - 2) - 1
17: 1 * ((4 + (3 * 5)) - 2)
18: 1 + ((4 + (3 * 5)) - 2)
19: (4 * (5 * (3 - 2))) - 1
20: 1 * (4 * (5 * (3 - 2)))
21: (2 * ((3 * 5) - 4)) - 1
22: 1 * (2 * ((3 * 5) - 4))
23: 1 + (2 * ((3 * 5) - 4))
24: ((3 * (4 + 5)) - 2) - 1
25: 1 * ((3 * (4 + 5)) - 2)
26: 1 + ((3 * (4 + 5)) - 2)
27: (4 * ((2 * 5) - 3)) - 1
28: (2 + (3 * (4 + 5))) - 1
29: 1 * (2 + (3 * (4 + 5)))
30: (3 + (4 * (2 + 5))) - 1
31: 1 * (3 + (4 * (2 + 5)))
32: 1 + (3 + (4 * (2 + 5)))
33: 1 * ((5 * (3 + 4)) - 2)
34: 1 + ((5 * (3 + 4)) - 2)
35: (3 * (4 * (5 - 2))) - 1
36: 1 * (3 * (4 * (5 - 2)))
37: 1 + (3 * (4 * (5 - 2)))
38: (3 * (5 + (2 * 4))) - 1
39: 1 * (3 * (5 + (2 * 4)))
40: 1 + (3 * (5 + (2 * 4)))
41: 1 + (4 * (2 + (3 + 5)))
42: 1 * (3 * (4 + (2 * 5)))
43: 1 + (3 * (4 + (2 * 5)))
44: (5 * (2 + (3 + 4))) - 1
45: (2 * (3 + (4 * 5))) - 1
46: 1 * (2 * (3 + (4 * 5)))
47: ((2 + 4) * (3 + 5)) - 1
48: 1 * ((2 + 4) * (3 + 5))
49: 1 + ((2 + 4) * (3 + 5))
50: 1 * (5 * ((3 * 4) - 2))
51: 1 + (5 * ((3 * 4) - 2))
52: 1 * (4 * ((3 * 5) - 2))
53: (2 * (3 * (4 + 5))) - 1
54: 1 * (2 * (3 * (4 + 5)))
55: 1 + (2 * (3 * (4 + 5)))
56: 1 + (5 * (3 + (2 * 4)))
57: ((3 * (4 * 5)) - 2) - 1
58: 1 * ((3 * (4 * 5)) - 2)
59: 1 - (2 - (3 * (4 * 5)))
60: 2 * (3 * (1 + (4 + 5)))
61: (2 + (3 * (4 * 5))) - 1
62: 1 * (2 + (3 * (4 * 5)))
63: 1 + (2 + (3 * (4 * 5)))
64: 1 * (2 * (4 * (3 + 5)))
65: 1 + (2 * (4 * (3 + 5)))
66: 1 * (3 * (2 + (4 * 5)))
67: 1 + (3 * (2 + (4 * 5)))
68: 1 * (4 * (2 + (3 * 5)))
69: (2 * (5 * (3 + 4))) - 1
70: 1 * (2 * (5 * (3 + 4)))
71: 1 + (2 * (5 * (3 + 4)))
72: 2 * (1 + (5 * (3 + 4)))
73: (3 * (5 * (1 + 4))) - 2
74: 2 + (3 * (4 * (1 + 5)))
75: 3 * (5 * ((2 + 4) - 1))

With exponents, we can go to 177:

76: 1 + (3 * (5 ^ (4 - 2)))
77: (2 - (5 - (3 ^ 4))) - 1
78: (4 + (3 * (5 ^ 2))) - 1
79: 1 * (4 + (3 * (5 ^ 2)))
80: (((2 ^ 3) - 5) ^ 4) - 1
81: 1 * (((2 ^ 3) - 5) ^ 4)
82: 1 + (((2 ^ 3) - 5) ^ 4)
83: ((5 + (3 ^ 4)) - 2) - 1
84: 1 * ((5 + (3 ^ 4)) - 2)
85: 1 + ((5 + (3 ^ 4)) - 2)
86: (3 * (4 + (5 ^ 2))) - 1
87: 1 * (3 * (4 + (5 ^ 2)))
88: 1 + (3 * (4 + (5 ^ 2)))
89: 1 + (2 + (5 + (3 ^ 4)))
90: 1 * (3 * (5 * (2 + 4)))
91: 1 + (3 * (5 * (2 + 4)))
92: 1 * ((3 * (2 ^ 5)) - 4)
93: 1 - (4 - (3 * (2 ^ 5)))
94: (5 * (3 + (2 ^ 4))) - 1
95: 1 * (5 * (3 + (2 ^ 4)))
96: ((4 * (5 ^ 2)) - 3) - 1
97: 1 * ((4 * (5 ^ 2)) - 3)
98: 1 + ((4 * (5 ^ 2)) - 3)
99: (4 * (5 * (2 + 3))) - 1
100: 1 * (4 * (5 * (2 + 3)))
101: 1 + (4 * (5 * (2 + 3)))
102: (3 + (4 * (5 ^ 2))) - 1
103: 1 * (3 + (4 * (5 ^ 2)))
104: 1 + (3 + (4 * (5 ^ 2)))
105: ((5 ^ 2) + (3 ^ 4)) - 1
106: 1 * ((5 ^ 2) + (3 ^ 4))
107: (3 * (4 + (2 ^ 5))) - 1
108: 1 / ((3 ^ (2 - 5)) / 4)
109: 1 + (3 * (4 + (2 ^ 5)))
110: 1 - ((2 ^ 4) - (5 ^ 3))
111: (4 * (3 + (5 ^ 2))) - 1
112: 1 * (4 * (3 + (5 ^ 2)))
113: 1 + (4 * (3 + (5 ^ 2)))
114: 1 + ((2 ^ 5) + (3 ^ 4))
115: (4 * ((2 ^ 5) - 3)) - 1
116: 1 * (4 * ((2 ^ 5) - 3))
117: (2 * ((4 ^ 3) - 5)) - 1
118: (((5 ^ 3) - 4) - 2) - 1
119: (2 * (3 * (4 * 5))) - 1
120: 1 * (2 * (3 * (4 * 5)))
121: 1 + (2 * (3 * (4 * 5)))
122: (2 - (4 - (5 ^ 3))) - 1
123: 1 * (2 - (4 - (5 ^ 3)))
124: 1 - ((4 - (5 ^ 3)) - 2)
125: 1 * ((4 * (2 ^ 5)) - 3)
126: ((4 + (5 ^ 3)) - 2) - 1
127: 1 * ((4 + (5 ^ 3)) - 2)
128: 1 - (2 - (4 + (5 ^ 3)))
129: 1 + (2 ^ ((3 * 4) - 5))
130: (2 + (4 + (5 ^ 3))) - 1
131: 1 * (2 + (4 + (5 ^ 3)))
132: 1 + (2 + (4 + (5 ^ 3)))
133: 1 * (5 + (2 ^ (3 + 4)))
134: 1 + (5 + (2 ^ (3 + 4)))
135: 3 * (5 * (1 + (2 * 4)))
136: 2 * ((5 + (4 ^ 3)) - 1)
137: (2 * (5 + (4 ^ 3))) - 1
138: 1 * (2 * (5 + (4 ^ 3)))
139: 1 + (2 * (5 + (4 ^ 3)))
140: 1 * (4 * (3 + (2 ^ 5)))
141: 1 + (4 * (3 + (2 ^ 5)))
142: 1 + ((2 ^ 4) + (5 ^ 3))
143: ((3 + (4 + 5)) ^ 2) - 1
144: 1 * ((3 + (4 + 5)) ^ 2)
145: 1 + ((3 + (4 + 5)) ^ 2)
146: (1 + (3 ^ (2 + 4))) / 5
147: 3 + (4 * ((1 + 5) ^ 2))
148: (5 + ((3 * 4) ^ 2)) - 1
149: 1 * (5 + ((3 * 4) ^ 2))
150: 1 + (5 + ((3 * 4) ^ 2))
151: (2 * ((3 ^ 4) - 5)) - 1
152: 1 * (2 * ((3 ^ 4) - 5))
153: 1 - (2 * (5 - (3 ^ 4)))
154: 2 * (1 - (5 - (3 ^ 4)))
155: 5 * (((4 ^ 3) / 2) - 1)
156: ((2 * (3 ^ 4)) - 5) - 1
157: 1 * ((2 * (3 ^ 4)) - 5)
158: 1 - (5 - (2 * (3 ^ 4)))
159: ((5 * (4 ^ 3)) / 2) - 1
160: 1 * ((5 * (4 ^ 3)) / 2)
161: 1 + ((5 * (4 ^ 3)) / 2)
162: 2 * (3 ^ (4 * (1 ^ 5)))
163: 3 + (5 * (2 ^ (1 + 4)))
164: 2 * ((1 ^ 5) + (3 ^ 4))
165: 5 * (1 + ((4 ^ 3) / 2))
166: (5 + (2 * (3 ^ 4))) - 1
167: 1 * (5 + (2 * (3 ^ 4)))
168: 1 + (5 + (2 * (3 ^ 4)))
169: (1 + (3 + (4 + 5))) ^ 2
170: 2 * ((5 + (3 ^ 4)) - 1)
171: (2 * (5 + (3 ^ 4))) - 1
172: 1 * (2 * (5 + (3 ^ 4)))
173: 1 + (2 * (5 + (3 ^ 4)))
174: ((3 + 4) * (5 ^ 2)) - 1
175: 1 * ((3 + 4) * (5 ^ 2))
176: 1 + ((3 + 4) * (5 ^ 2))
177: 3 * ((4 ^ (1 + 2)) - 5)

Out the first 500 (nonzero) natural numbers, we can make 395. Out of the first 1000, we can make 554.

Bonus: including integer division only takes us a bit further, to 182.

178: (5 ^ 4) // (3 + (1 / 2))
179: (4 * (5 * (3 ^ 2))) - 1
180: ((2 + (5 / 3)) ^ 4) // 1
181: 1 + (4 * (5 * (3 ^ 2)))
182: (5 ^ (4 - (1 / 3))) // 2

I added flags for modulo and bitwise and/or/xor too, but those don't extend our reach at all beyond 182. Out of binary operations for now...

Bonus 2: When we add concatenation to the original four + exponentiation, it takes us all the way to 668, and covers 980 of the first 1000 numbers!

178: 1 || (2 * (3 || (4 + 5)))
179: 1 || (4 + (3 * (5 ^ 2)))
180: 1 * (4 * (5 * (3 ^ 2)))
181: (2 + (4 ^ (5 - 3))) || 1
182: 1 || ((4 * (5 - 3)) || 2)
183: 1 || (((2 ^ 5) / 4) || 3)
184: 1 || (2 * ((4 || 5) - 3))
185: 1 || ((5 * (3 || 4)) / 2)
186: 1 || (2 + ((3 + 5) || 4))
187: 1 || (3 * (2 || (4 + 5)))
188: 1 || (2 + (5 + (3 ^ 4)))
189: 1 || (((4 + 5) || 2) - 3)
190: 1 || (3 * (5 * (2 + 4)))
191: (2 + ((4 * 5) - 3)) || 1
192: 1 || ((3 * (2 ^ 5)) - 4)
193: 1 || (3 + (2 * (4 || 5)))
194: 1 || ((3 * (5 - 2)) || 4)
195: 1 || (2 + ((4 + 5) || 3))
196: 1 || (2 * (3 + (4 || 5)))
197: 1 || ((4 * (5 ^ 2)) - 3)
198: (((4 * 5) || 2) - 3) - 1
199: 1 || (((2 * 5) || 3) - 4)
200: (((4 * 5) || 3) - 2) - 1
201: (2 + ((5 || 4) / 3)) || 1
202: 1 + (((4 * 5) || 3) - 2)
203: (4 * ((5 || 3) - 2)) - 1
204: 1 * (4 * ((5 || 3) - 2))
205: 1 + (4 * ((5 || 3) - 2))
206: ((5 * (4 || 2)) - 3) - 1
207: 1 * ((5 * (4 || 2)) - 3)
208: ((2 + (5 ^ 4)) / 3) - 1
209: 1 * ((2 + (5 ^ 4)) / 3)
210: (2 || ((3 * 5) - 4)) - 1
211: (2 || ((5 - 4) ^ 3)) || 1
212: 1 + (2 || ((3 * 5) - 4))
213: 1 * (2 || ((5 - 4) || 3))
214: 1 + (2 || ((5 - 4) || 3))
215: (2 || (4 ^ (5 - 3))) - 1
216: 1 * (2 || (4 ^ (5 - 3)))
217: 1 + (2 || (4 ^ (5 - 3)))
218: (2 || (4 + (3 * 5))) - 1
219: 1 * (2 || (4 + (3 * 5)))
220: (((3 * 5) ^ 2) - 4) - 1
221: (2 || (3 + (4 - 5))) || 1
222: 1 - (4 - ((3 * 5) ^ 2))
223: (2 || ((5 - 3) || 4)) - 1
224: 1 * (2 || ((5 - 3) || 4))
225: 1 - (((3 - 5) || 2) || 4)
226: 1 + (((4 || 5) / 3) ^ 2)
227: 1 * (2 || (3 * (4 + 5)))
228: (2 || ((3 || 4) - 5)) - 1
229: (5 * (4 || (2 * 3))) - 1
230: (2 || (3 || (5 - 4))) - 1
231: (2 || (3 * (5 - 4))) || 1
232: 1 + (2 || (3 || (5 - 4)))
233: 1 + ((3 + (4 * 5)) || 2)
234: (2 || (5 * (3 + 4))) - 1
235: 1 * (2 || (5 * (3 + 4)))
236: 1 + (2 || (5 * (3 + 4)))
237: 1 + (4 * (5 || (3 ^ 2)))
238: 1 * (2 || ((4 || 3) - 5))
239: (3 * (5 * (2 ^ 4))) - 1
240: (2 - (4 - (3 ^ 5))) - 1
241: (2 || (3 - (4 - 5))) || 1
242: 1 + (2 - (4 - (3 ^ 5)))
243: 1 + (2 * ((5 ^ 3) - 4))
244: 1 + (3 * ((4 + 5) ^ 2))
245: ((2 * (5 ^ 3)) - 4) - 1
246: 1 * ((2 * (5 ^ 3)) - 4)
247: 1 + ((2 * (5 ^ 3)) - 4)
248: (2 || ((5 || 3) - 4)) - 1
249: 1 * (2 || ((5 || 3) - 4))
250: (2 || ((5 || 4) - 3)) - 1
251: ((5 * (4 - 3)) ^ 2) || 1
252: 1 + (2 || ((5 || 4) - 3))
253: ((5 * (2 + 3)) || 4) - 1
254: (3 * ((2 * 4) || 5)) - 1
255: 1 * (3 * ((2 * 4) || 5))
256: (2 || (3 + (5 || 4))) - 1
257: 1 * (2 || (3 + (5 || 4)))
258: 1 + (2 || (3 + (5 || 4)))
259: 1 + (2 * (4 + (5 ^ 3)))
260: 1 * (2 || (3 * (4 * 5)))
261: (2 * ((5 - 4) || 3)) || 1
262: 2 * (((5 - 4) || 3) || 1)
263: 2 || (3 * (1 + (4 * 5)))
264: 2 * (1 || (4 * (3 + 5)))
265: 2 || (5 * (1 + (3 * 4)))
266: (((5 || 3) || 4) / 2) - 1
267: 1 / (2 / ((5 || 3) || 4))
268: 1 + (((5 || 3) || 4) / 2)
269: (2 * (3 * (4 || 5))) - 1
270: 1 * (2 * (3 * (4 || 5)))
271: (2 || ((3 * 4) - 5)) || 1
272: 1 * ((3 * (4 + 5)) || 2)
273: 1 + ((3 * (4 + 5)) || 2)
274: (2 || ((3 + 4) || 5)) - 1
275: 1 * (2 || ((3 + 4) || 5))
276: 1 + (2 || ((3 + 4) || 5))
277: 1 + (2 || ((3 ^ 4) - 5))
278: 2 * (1 || (3 || (4 + 5)))
279: (2 * (4 * (3 || 5))) - 1
280: 1 * (2 * (4 * (3 || 5)))
281: (2 || (4 * (5 - 3))) || 1
282: 1 * ((5 || (4 ^ 3)) / 2)
283: 1 + ((5 || (4 ^ 3)) / 2)
284: 1 * (2 || ((3 + 5) || 4))
285: (2 || (5 + (3 ^ 4))) - 1
286: 1 * (2 || (5 + (3 ^ 4)))
287: 1 + (2 || (5 + (3 ^ 4)))
288: (((4 * 5) - 3) ^ 2) - 1
289: 1 * (((4 * 5) - 3) ^ 2)
290: 1 + (((4 * 5) - 3) ^ 2)
291: ((3 || (5 - 4)) - 2) || 1
292: 1 * (((3 || 4) - 5) || 2)
293: 1 - ((5 - (3 || 4)) || 2)
294: 1 + (2 || ((4 + 5) || 3))
295: 1 - ((3 - (2 ^ 5)) || 4)
296: 2 * (1 || (3 + (4 || 5)))
297: ((5 ^ 4) - (3 || 1)) / 2
298: 2 * (1 || ((5 || 3) - 4))
299: (3 * (4 * (5 ^ 2))) - 1
300: ((2 + 5) * (4 || 3)) - 1
301: (2 * ((4 || 5) / 3)) || 1
302: 1 + ((2 + 5) * (4 || 3))
303: 1 * ((5 * (2 + 4)) || 3)
304: 1 + ((5 * (2 + 4)) || 3)
305: ((3 || (2 * 5)) - 4) - 1
306: 1 * ((3 || (2 * 5)) - 4)
307: 1 + ((3 || (2 * 5)) - 4)
308: 1 * (2 * ((3 * 5) || 4))
309: (3 || ((4 * 5) / 2)) - 1
310: (((5 ^ 4) - 3) / 2) - 1
311: (2 - (5 - (3 || 4))) || 1
312: 1 + (((5 ^ 4) - 3) / 2)
313: 1 + ((3 || (5 - 4)) || 2)
314: (5 * ((2 + 4) || 3)) - 1
315: 1 * (5 * ((2 + 4) || 3))
316: 1 + (5 * ((2 + 4) || 3))
317: ((3 || (4 * 5)) - 2) - 1
318: (3 || ((2 || 4) - 5)) - 1
319: 1 * (3 || ((2 || 4) - 5))
320: (3 || (2 || (5 - 4))) - 1
321: ((3 * (5 - 4)) || 2) || 1
322: 1 + (3 || (2 || (5 - 4)))
323: (((5 || 4) / 3) ^ 2) - 1
324: 1 * (((5 || 4) / 3) ^ 2)
325: 1 + (((5 || 4) / 3) ^ 2)
326: 1 + (3 || (5 ^ (4 - 2)))
327: (3 || ((2 ^ 5) - 4)) - 1
328: 1 * (3 || ((2 ^ 5) - 4))
329: 1 + (3 || ((2 ^ 5) - 4))
330: 1 * (3 || (5 * (2 + 4)))
331: (2 + (3 || (5 - 4))) || 1
332: 1 * (3 || (4 ^ (5 / 2)))
333: 1 + (3 || (4 ^ (5 / 2)))
334: 1 * (3 || ((5 - 2) || 4))
335: 1 + (3 || ((5 - 2) || 4))
336: 1 * (3 || (4 + (2 ^ 5)))
337: 1 + (3 || (4 + (2 ^ 5)))
338: (((2 + 5) ^ 3) - 4) - 1
339: 1 * (((2 + 5) ^ 3) - 4)
340: 1 - (4 - ((2 + 5) ^ 3))
341: (2 + (4 * (3 + 5))) || 1
342: ((3 || (4 || 5)) - 2) - 1
343: 1 * ((3 || (4 || 5)) - 2)
344: 1 + ((3 || (4 || 5)) - 2)
345: (3 || ((5 || 1) - 4)) - 2
346: (2 + (3 || (4 || 5))) - 1
347: 1 * (2 + (3 || (4 || 5)))
348: 1 + (2 + (3 || (4 || 5)))
349: 1 + (3 || ((5 || 2) - 4))
350: 2 * (1 || ((3 + 4) || 5))
351: (3 + (4 ^ (5 / 2))) || 1
352: 1 * ((5 * (3 + 4)) || 2)
353: 1 + ((5 * (3 + 4)) || 2)
354: 1 * ((3 + (2 ^ 5)) || 4)
355: 1 + ((3 + (2 ^ 5)) || 4)
356: 1 * (2 + (3 || (5 || 4)))
357: 1 + (2 + (3 || (5 || 4)))
358: 1 * (3 || (5 || (2 * 4)))
359: (3 * (5 * (2 || 4))) - 1
360: ((4 + (3 * 5)) ^ 2) - 1
361: (((4 + 5) - 3) ^ 2) || 1
362: 1 + ((4 + (3 * 5)) ^ 2)
363: (3 || (4 ^ (5 - 2))) - 1
364: (3 || ((2 + 4) || 5)) - 1
365: 1 * (3 || ((2 + 4) || 5))
366: 1 + (3 || ((2 + 4) || 5))
367: 3 || (((2 + 5) || 1) - 4)
368: 2 * (1 || ((3 + 5) || 4))
369: 3 * ((5 ^ (4 - 1)) - 2)
370: 3 || (2 * ((4 - 1) || 5))
371: (2 + (5 * (3 + 4))) || 1
372: (((4 || 2) - 5) || 3) - 1
373: 1 * (((4 || 2) - 5) || 3)
374: 1 + (((4 || 2) - 5) || 3)
375: 1 + (3 || ((2 + 5) || 4))
376: (1 || (4 * (2 ^ 5))) / 3
377: 2 + (3 * (5 ^ (4 - 1)))
378: ((2 || 1) * (5 || 4)) / 3
379: (3 || (5 * (2 ^ 4))) - 1
380: (3 || ((4 + 5) ^ 2)) - 1
381: (2 * (4 + (3 * 5))) || 1
382: 1 - ((2 - 5) || (3 ^ 4))
383: 1 - ((5 - (4 || 3)) || 2)
384: 1 * (3 * (4 * (2 ^ 5)))
385: 1 + (3 * (4 * (2 ^ 5)))
386: 1 + (3 || ((2 * 4) || 5))
387: 2 + (((4 || 1) - 3) || 5)
388: 3 || (2 * ((4 || 5) - 1))
389: (3 || (2 * (4 || 5))) - 1
390: 1 * (3 || (2 * (4 || 5)))
391: (3 * ((5 || 2) / 4)) || 1
392: 1 * ((3 || (4 + 5)) || 2)
393: 1 + ((3 || (4 + 5)) || 2)
394: (((4 || 2) - 3) || 5) - 1
395: 1 * (((4 || 2) - 3) || 5)
396: (((4 * 5) ^ 2) - 3) - 1
397: 1 * (((4 * 5) ^ 2) - 3)
398: 1 - (3 - ((4 * 5) ^ 2))
399: 3 || ((4 * (5 ^ 2)) - 1)
400: (5 / ((3 / 4) - 1)) ^ 2
401: (((4 || 5) - 3) - 2) || 1
402: ((5 * (3 ^ 4)) - 2) - 1
403: ((5 * (2 ^ 3)) || 4) - 1
404: 1 * ((5 * (2 ^ 3)) || 4)
405: (2 * ((4 * 5) || 3)) - 1
406: 1 * (2 * ((4 * 5) || 3))
407: 1 + (2 * ((4 * 5) || 3))
408: 1 + (2 + (5 * (3 ^ 4)))
409: (4 || (2 + (3 + 5))) - 1
410: (4 || (5 + (2 * 3))) - 1
411: (2 + (3 || (4 + 5))) || 1
412: 1 + (4 || (5 + (2 * 3)))
413: (4 || (5 + (3 ^ 2))) - 1
414: 1 * (4 || (5 + (3 ^ 2)))
415: (4 || (2 * (3 + 5))) - 1
416: 1 * (4 || (2 * (3 + 5)))
417: 1 + (4 || (2 * (3 + 5)))
418: 1 + (2 + (4 || (3 * 5)))
419: 1 - (5 - ((4 || 2) || 3))
420: (4 || (3 * (2 + 5))) - 1
421: ((3 - (4 - 5)) || 2) || 1
422: 1 + (4 || (3 * (2 + 5)))
423: (2 * (4 * (5 || 3))) - 1
424: 1 * (2 * (4 * (5 || 3)))
425: 1 + (2 * (4 * (5 || 3)))
426: 1 + (4 || (5 * (2 + 3)))
427: (3 + (4 || (5 ^ 2))) - 1
428: ((4 || (2 ^ 5)) - 3) - 1
429: 1 * ((4 || (2 ^ 5)) - 3)
430: 1 - (3 - (4 || (2 ^ 5)))
431: ((5 + (3 ^ 4)) / 2) || 1
432: ((4 || (3 || 5)) - 2) - 1
433: 1 * ((4 || (3 || 5)) - 2)
434: 1 + ((4 || (3 || 5)) - 2)
435: 1 * (3 + (4 || (2 ^ 5)))
436: 1 + (3 + (4 || (2 ^ 5)))
437: 1 * (2 + (4 || (3 || 5)))
438: 1 + (2 + (4 || (3 || 5)))
439: (4 || (5 * (2 ^ 3))) - 1
440: 1 * (4 || (5 * (2 ^ 3)))
441: (2 + ((4 || 5) - 3)) || 1
442: (3 + (4 || (1 ^ 5))) || 2
443: (4 || (3 * (1 || 5))) - 2
444: (4 || (5 * (3 ^ 2))) - 1
445: 1 * (4 || (5 * (3 ^ 2)))
446: 1 + (4 || (5 * (3 ^ 2)))
447: ((2 + 5) * (4 ^ 3)) - 1
448: (((4 || 5) || 2) - 3) - 1
449: 1 * (((4 || 5) || 2) - 3)
450: (((4 || 5) || 3) - 2) - 1
451: (((5 || 2) - 4) - 3) || 1
452: 1 + (((4 || 5) || 3) - 2)
453: ((5 * (3 ^ 2)) || 4) - 1
454: 1 * ((5 * (3 ^ 2)) || 4)
455: 1 + ((5 * (3 ^ 2)) || 4)
456: 1 + (2 + ((4 || 5) || 3))
457: 1 + (4 || (5 || (2 * 3)))
458: 1 * (4 || (5 || (2 ^ 3)))
459: (4 * (5 * (2 || 3))) - 1
460: 1 * (4 * (5 * (2 || 3)))
461: (2 * (3 + (4 * 5))) || 1
462: 2 * ((3 + (4 * 5)) || 1)
463: (4 || ((3 + 5) ^ 2)) - 1
464: (4 || ((2 * 3) || 5)) - 1
465: 1 * (4 || ((2 * 3) || 5))
466: 1 + (4 || ((2 * 3) || 5))
467: 2 + (4 || (5 * (1 || 3)))
468: ((2 + (4 || 5)) || 1) - 3
469: (4 || (2 * (3 || 5))) - 1
470: 1 * (4 || (2 * (3 || 5)))
471: (((5 || 3) - 4) - 2) || 1
472: ((2 + (4 || 5)) || 3) - 1
473: 1 * ((2 + (4 || 5)) || 3)
474: 1 + ((2 + (4 || 5)) || 3)
475: 1 * (4 || (3 * (5 ^ 2)))
476: 1 + (4 || (3 * (5 ^ 2)))
477: (2 * ((3 ^ 5) - 4)) - 1
478: 1 * (2 * ((3 ^ 5) - 4))
479: 1 - (2 * (4 - (3 ^ 5)))
480: 2 * (1 - (4 - (3 ^ 5)))
481: (2 * ((5 - 3) || 4)) || 1
482: 1 * ((3 + (4 || 5)) || 2)
483: 1 - (4 - (2 * (3 ^ 5)))
484: (4 || ((2 ^ 3) || 5)) - 1
485: 1 * (4 || ((2 ^ 3) || 5))
486: 1 + (4 || ((2 ^ 3) || 5))
487: 1 + ((4 * (3 ^ 5)) / 2)
488: 2 * ((1 ^ 4) + (3 ^ 5))
489: (4 + (2 * (3 ^ 5))) - 1
490: 1 * (4 + (2 * (3 ^ 5)))
491: (((5 || 4) - 3) - 2) || 1
492: 1 * (((5 || 3) - 4) || 2)
493: 1 - ((4 - (5 || 3)) || 2)
494: 1 * (2 * (4 + (3 ^ 5)))
495: 1 + (2 * (4 + (3 ^ 5)))
496: 1 + (3 * ((2 ^ 4) || 5))
497: ((4 * (5 ^ 3)) - 2) - 1
498: 1 * ((4 * (5 ^ 3)) - 2)
499: 1 - (2 - (4 * (5 ^ 3)))
500: ((1 + (4 + 5)) ^ 3) / 2
501: (2 + (3 + (4 || 5))) || 1
502: 1 * (2 + (4 * (5 ^ 3)))
503: 1 + (2 + (4 * (5 ^ 3)))
504: 2 * (4 * ((1 + 5) || 3))
505: (((5 || 1) || 2) - 4) - 3
506: (((2 * 4) ^ 3) - 5) - 1
507: (4 * (2 + (5 ^ 3))) - 1
508: ((2 ^ (4 + 5)) - 3) - 1
509: ((5 || (3 * 4)) - 2) - 1
510: (5 || (3 + (2 * 4))) - 1
511: (2 - (4 - (5 || 3))) || 1
512: 1 + (5 || (3 + (2 * 4)))
513: 1 * ((5 || (2 ^ 4)) - 3)
514: 1 + ((5 || (2 ^ 4)) - 3)
515: 1 * (3 + (2 ^ (4 + 5)))
516: (5 || ((3 || 4) / 2)) - 1
517: 1 * (5 || ((3 || 4) / 2))
518: 1 + (5 || ((3 || 4) / 2))
519: 1 + (5 || (3 * (2 + 4)))
520: (((5 || 2) || 4) - 3) - 1
521: ((5 * (4 - 3)) || 2) || 1
522: 1 + (((5 || 2) || 4) - 3)
523: (5 || (2 * (3 * 4))) - 1
524: 1 * (5 || (2 * (3 * 4)))
525: 1 + (5 || (2 * (3 * 4)))
526: (3 + ((5 || 2) || 4)) - 1
527: 1 * (3 + ((5 || 2) || 4))
528: ((3 + (4 * 5)) ^ 2) - 1
529: 1 * ((3 + (4 * 5)) ^ 2)
530: 1 + ((3 + (4 * 5)) ^ 2)
531: (2 + ((5 || 4) - 3)) || 1
532: 1 * (((5 || 3) || 4) - 2)
533: 1 + (((5 || 3) || 4) - 2)
534: (2 + (5 || (1 ^ 3))) || 4
535: (2 + ((5 || 3) || 4)) - 1
536: 1 * (2 + ((5 || 3) || 4))
537: 1 + (2 + ((5 || 3) || 4))
538: (((5 || 4) || 2) - 3) - 1
539: 1 * (((5 || 4) || 2) - 3)
540: (((5 || 4) || 3) - 2) - 1
541: (2 * (3 * (4 + 5))) || 1
542: 1 + (((5 || 4) || 3) - 2)
543: 2 + (5 || (4 || (1 ^ 3)))
544: (2 + ((5 || 4) || 3)) - 1
545: 1 * (2 + ((5 || 4) || 3))
546: 1 + (2 + ((5 || 4) || 3))
547: 1 + (5 || (4 || (2 * 3)))
548: (5 || ((3 + 4) ^ 2)) - 1
549: 1 * (5 || ((3 + 4) ^ 2))
550: 1 + (5 || ((3 + 4) ^ 2))
551: ((3 + (5 || 4)) - 2) || 1
552: (4 + (5 || (1 ^ 3))) || 2
553: ((2 + (5 || 3)) || 4) - 1
554: 1 * ((2 + (5 || 3)) || 4)
555: 1 + ((2 + (5 || 3)) || 4)
556: 5 || ((1 || (3 * 4)) / 2)
557: ((5 || (2 * 3)) || 1) - 4
558: (5 * (1 || (3 * 4))) - 2
559: ((2 ^ 4) * (3 || 5)) - 1
560: 1 * ((2 ^ 4) * (3 || 5))
561: (4 * (5 + (3 ^ 2))) || 1
562: ((2 + (5 || 4)) || 3) - 1
563: 1 * ((2 + (5 || 4)) || 3)
564: 1 + ((2 + (5 || 4)) || 3)
565: 1 + ((5 || (2 * 3)) || 4)
566: 1 * (2 + (5 || (4 ^ 3)))
567: 1 + (2 + (5 || (4 ^ 3)))
568: 1 * (5 || (2 * (3 || 4)))
569: 1 + (5 || (2 * (3 || 4)))
570: (1 || (4 * (3 || 5))) / 2
571: (((4 ^ 3) - 5) - 2) || 1
572: 1 * ((3 + (5 || 4)) || 2)
573: 1 + ((3 + (5 || 4)) || 2)
574: (1 + (5 || (2 * 3))) || 4
575: (((5 - 3) || 4) ^ 2) - 1
576: 1 * (((5 - 3) || 4) ^ 2)
577: 1 + (((5 - 3) || 4) ^ 2)
578: ((5 || (3 ^ 4)) - 2) - 1
579: 1 * ((5 || (3 ^ 4)) - 2)
580: 1 - (2 - (5 || (3 ^ 4)))
581: (2 * ((3 || 4) - 5)) || 1
582: (2 + (5 || (3 ^ 4))) - 1
583: 1 * (2 + (5 || (3 ^ 4)))
584: 1 + (2 + (5 || (3 ^ 4)))
585: 1 + ((5 || (2 ^ 3)) || 4)
586: 1 * (5 || (2 * (4 || 3)))
587: 1 + (5 || (2 * (4 || 3)))
588: 3 * (((1 ^ 5) || 4) ^ 2)
589: (((4 ^ 3) - 5) || 1) - 2
590: 5 * (1 || (3 * (2 + 4)))
591: (2 + (3 + (5 || 4))) || 1
592: 1 * (((4 ^ 3) - 5) || 2)
593: 1 + (((4 ^ 3) - 5) || 2)
594: 1 * ((5 || (3 ^ 2)) || 4)
595: 1 + ((5 || (3 ^ 2)) || 4)
596: 2 + ((5 ^ 4) - (3 || 1))
597: 3 * (((4 * 5) || 1) - 2)
598: (5 ^ 4) - (3 ^ (1 + 2))
599: ((3 * (4 * 5)) || 1) - 2
600: 5 * (1 || (4 * (2 + 3)))
601: (4 * ((3 - 2) || 5)) || 1
602: 1 * ((3 * (4 * 5)) || 2)
603: 1 + ((3 * (4 * 5)) || 2)
604: 4 * (((3 - 2) || 5) || 1)
605: (3 * ((4 * 5) || 2)) - 1
606: 1 * (3 * ((4 * 5) || 2))
607: 1 + (3 * ((4 * 5) || 2))
608: 1 * (4 * ((3 * 5) || 2))
609: 1 + (4 * ((3 * 5) || 2))
610: 1 * (5 * ((3 * 4) || 2))
611: (2 + ((4 ^ 3) - 5)) || 1
612: (((4 + 5) - 3) || 1) || 2
613: (3 * (5 * (4 || 1))) - 2
614: ((2 + 4) || (3 * 5)) - 1
615: 1 * ((2 + 4) || (3 * 5))
616: 1 + ((2 + 4) || (3 * 5))
617: 1 + ((5 ^ 4) - (3 ^ 2))
618: ((5 ^ 4) - (2 * 3)) - 1
619: (((5 ^ 4) - 3) - 2) - 1
620: 1 * (((5 ^ 4) - 3) - 2)
621: (2 * (3 || (5 - 4))) || 1
622: 2 * ((3 || (5 - 4)) || 1)
623: (2 - (3 - (5 ^ 4))) - 1
624: 1 * (2 - (3 - (5 ^ 4)))
625: ((3 + (5 ^ 4)) - 2) - 1
626: 1 * ((3 + (5 ^ 4)) - 2)
627: 1 + ((3 + (5 ^ 4)) - 2)
628: 1 + (((5 ^ 3) || 4) / 2)
629: (2 + (3 + (5 ^ 4))) - 1
630: ((2 * 3) + (5 ^ 4)) - 1
631: (3 * (2 || (5 - 4))) || 1
632: 1 + ((2 * 3) + (5 ^ 4))
633: ((3 ^ 2) + (5 ^ 4)) - 1
634: 1 * ((3 ^ 2) + (5 ^ 4))
635: 1 + ((3 ^ 2) + (5 ^ 4))
636: 1 + (((2 + 4) || 3) || 5)
637: 1 * (((4 ^ 3) || 2) - 5)
638: 1 - (5 - ((4 ^ 3) || 2))
639: (2 * (3 || (4 * 5))) - 1
640: 1 * (2 * (3 || (4 * 5)))
641: (2 ^ ((4 + 5) - 3)) || 1
642: (((4 ^ 3) || 5) - 2) - 1
643: 1 * (((4 ^ 3) || 5) - 2)
644: 1 + (((4 ^ 3) || 5) - 2)
645: 1 + (((3 + 5) ^ 2) || 4)
646: 1 + (((2 * 3) || 4) || 5)
647: ((2 || 3) + (5 ^ 4)) - 1
648: 1 * ((2 || 3) + (5 ^ 4))
649: 1 + ((2 || 3) + (5 ^ 4))
650: 2 * (3 || (5 * (1 + 4)))
651: (5 * ((2 ^ 4) - 3)) || 1
652: (((2 + 4) || 5) || 3) - 1
653: 1 * (((2 + 4) || 5) || 3)
654: 1 + (((2 + 4) || 5) || 3)
655: 1 + (((2 * 3) || 5) || 4)
656: ((3 || 2) + (5 ^ 4)) - 1
657: 1 * ((3 || 2) + (5 ^ 4))
658: 1 + ((3 || 2) + (5 ^ 4))
659: (3 * (2 || (4 * 5))) - 1
660: 1 * (3 * (2 || (4 * 5)))
661: (3 * (2 + (4 * 5))) || 1
662: (((3 + 4) || 1) - 5) || 2
663: 3 * ((2 + (4 * 5)) || 1)
664: ((2 + (4 ^ 3)) || 5) - 1
665: 1 * ((2 + (4 ^ 3)) || 5)
666: 1 + ((2 + (4 ^ 3)) || 5)
667: (4 || 2) + (5 ^ (1 + 3))
668: (5 * (1 || (3 || 4))) - 2

Bonus 3: If we extend this to include 6 (output), we cover 9965 of the first 10000, the first one we fail to make isn't until 5989.

Answer 3

"a guy" has covered the initial question here is the bonus

$76 = 3^4 - 5$
$77 = 3^4 + 1 - 5$
$78 = 3^4 + 2 - 5$
$79 = 3^4 - 2$
$80 = 3^4 - 1$
$81 = 3^4$
$82 = 3^4 + 1$
$83 = 3^4 + 2$
$84 = 3^4 + 2 + 1$
$85 = 3^4 + 5 - 1$
$86 = 3^4 + 5$
$87 = 3^4 + 5 + 1$
$88 = 3^4 + 5 + 2$
$89 = 3^4 + 5 + 2 + 1$
$90 = 3^4 + (5\times 2) - 1$
$91 = 3^4 + (5\times 2)$
$92 = 3^4 + (5\times 2) + 1$
$93 = 3^4 + ((5+1)\times 2)$
$94 = ((2^4 + 3) \times 5) - 1$
$95 = ((2^4 + 3) \times 5)$
$96 = ((2^4 + 3) \times 5) + 1$
$97 = (2^5 \times 3) + 1$
$98 = (5^2 \times 4) - 3 + 1$
$99 = (5^2 \times 4) - 1$
$100 = (5^2 \times 4)$
$101 = (5^2 \times 4) + 1$
$102 = (5^2 \times 4) + 3 - 1$
$103 = (5^2 \times 4) + 3$
$104 = (5^2 \times 4) + 3 + 1$
$105 = 3 \times 5 \times (1+2+4)$
$106 = 3^4 + 5^2$
$107 = 3^4 + 5^2 + 1$
$108 = (5^2 + 3 - 1) \times 4$
$109 = ((2^5 + 4) \times 3) + 1$
$110 = ((4\times 3) - 1) \times 2 \times 5$
$111 = ((5^2 + 3) \times 4) - 1$
$112 = ((5^2 + 3) \times 4)$
$113 = ((5^2 + 3) \times 4) + 1$
$114 = ((5\times 4)-1) \times 3 \times 2$
$115 = 5^3 - ((4+1)\times 2)$
$116 = 5^3 - (4\times 2) - 1$
$117 = 5^3 - (4\times 2)$
$118 = 5^3 - (4\times 2) + 1$
$119 = 5^3 - 4 - 2$
$120 = 5^3 - 4 - 1$
$121 = 5^3 - 4$
$122 = 5^3 - 2 - 1$
$123 = 5^3 - 2$
$124 = 5^3 - 1$
$125 = 5^3$

We can even go a bit further, without much difficulty

$126 = 5^3 + 1$
$127 = 5^3 + 2$
$128 = 5^3 + 2 + 1$
$129 = 5^3 + 4$
$130 = 5^3 + 4 + 1$
$131 = 5^3 + 4 + 2$
$132 = 5^3 + 4 + 2 + 1$
$133 = 5^3 + (4\times 2)$
$134 = 5^3 + (4\times 2) + 1$
$135 = 5^3 + ((4+1) \times 2)$

Here are a few more

$136 = (2^5 + 3 - 1) \times 4$
$137 = 5^3 + (4 \times (2+1))$
$138 = (4^3 + 5) \times 2$
$139 = ((4^3 + 5) \times 2) + 1$
$140 = (4^3 + 5 + 1) \times 2$
$141 = 5^3 + 2^4$
$142 = 5^3 + 2^4 + 1$
$143 = (3 \times 4)^2 - 1$
$144 = (3 \times 4)^2$
$145 = (3 \times 4)^2 + 1$