# A way to express 100 by using the first four natural numbers [closed]

Is there a way to express 100 by using the first four natural numbers in order?

The numbers 1, 2, 3, 4 can be linked by using $$+$$, $$-$$, $$\times$$, $$\div$$, $$($$ $$)$$, $$!$$ and exponents are also allowed. But you are not allowed to leave the integers.

Examples: $$-1+(2+3)^4=624$$ $$(1!+2)^{3^4}= large$$

• Brute force is likely applicable for this kind of problem, just list all possibilities using a computer...
– PC1
Nov 17, 2021 at 17:17
• Can you explain your first example? I evaluate the left hand side to be $624$. Also, what is the : operator? Nov 17, 2021 at 17:21
• @PC1 you are absolutly right. Nov 17, 2021 at 17:23
• Is concatenation accepted? That is, would $12+3^4=93$ be a valid expression? Nov 18, 2021 at 3:06
• $(1+(-2+3!)!)\cdot 4=100$ Nov 18, 2021 at 6:53

Seems more like a Puzzling.SE question to me but here's a CW post for others to edit. 79/100 complete

Number Expression
1 $$-1+2\cdot3-4$$
2 $$1+2+3-4$$
3 $$1+2\cdot3-4$$
4 $$1+2-3+4$$
5 $$1^{2\cdot3}+4$$
6 $$1-2+3+4$$
7 $$-1+2\cdot3!-4$$
8 $$-1+2+3+4$$
9 $$1\cdot2+3+4$$
10 $$1+2+3+4$$
11 $$1-2+3\cdot4$$
12 $$1^2\cdot3\cdot4$$
13 $$1+2+3!+4$$
14 $$1\cdot(2+3\cdot4)$$
15 $$-(1+2)!-3+4!$$
16 $$-1\cdot2^3+4!$$
17 $$1-2^3+4!$$
18 $$-1\cdot2\cdot3+4!$$
19 $$-1 + (2+3)\cdot 4$$
20 $$(-1+2+3)!-4$$
21 $$-(1+2)!+3+4!$$
22 $$-(-1^2+3)!+4!$$
23 $$-(1^{2\cdot3})+4!$$
24 $$-1-2+3+4!$$
25 $$-(1\cdot2)+3+4!$$
26 $$1-2+3+4!$$
27 $$-1-2+3!+4!$$
28 $$-1+2+3+4!$$
29 $$1-2+3!+4!$$
30 $$1+2+3+4!$$
31 $$-1+2+3!+4!$$
32 $$1\cdot2^3+4!$$
33 $$1+2+3!+4!$$
34
35 $$-1+2!\cdot3!+4!$$
36 $$1\cdot2!\cdot3!+4!$$
37 $$1+2!\cdot3!+4!$$
38
39 $$-1+2^{3!}-4!$$
40 $$(1+2)! \cdot 3! +4$$
41 $$1+2^{3!}-4!$$
42 $$(1+2)!\cdot(3+4)$$
43 $$1+2\cdot(-3+4!)$$
44 $$(-1+2\cdot3!)\cdot4$$
45
46
47 $$-1+2\cdot3!\cdot4$$
48 $$(1^2+3)!+4!$$
49 $$1+2\cdot3!\cdot4$$
50
51 $$(1+2)^3+4!$$
52 $$(1+2\cdot3!)\cdot4$$
53 $$-1+2\cdot(3+4!)$$
54 $$(1+2)!^3/4$$
55 $$1+2\cdot(3+4!)$$
56
57
58
59 $$-1+2^{3!}-4$$
60 $$(1+2)!\cdot3!+4!$$
61 $$1+2^{3!}-4$$
62
63 $$(1+2)\cdot(-3+4!)$$
64
65
66
67 $$-1+2^{3!}+4$$
68 $$1\cdot2^{3!}+4$$
69 $$1+2^{3!}+4$$
70 $$-1\cdot2+3\cdot4!$$
71 $$-1^2+3\cdot4!$$
72 $$1^2\cdot3\cdot4!$$
73 $$1^2+3\cdot4!$$
74 $$1\cdot2+3\cdot4!$$
75 $$-(1+2)!+3^4$$
76
77
78 $$-1-2+3^4$$
79 $$-(1\cdot2)+3^4$$
80 $$1-2+3^4$$
81 $$1^2\cdot3^4$$
82 $$-1+2+3^4$$
83 $$1\cdot2+3^4$$
84 $$1+2+3^4$$
85
86
87 $$-1+2^{3!}+4!$$
88 $$1\cdot(2^{3!}+4!)$$
89 $$1+2^{3!}+4!$$
90 $$(1+2)\cdot(3!)!/4!$$
91
92 $$(1-(-2+3!)!)\cdot(-4)$$
93
94
95 $$-1+(2+3)!-4!$$
96 $$1\cdot(2+3)!-4!$$
97 $$1+(2+3)!-4!$$
98
99
100 $$(1+(-2+3!)!)\cdot 4$$