# Create all numbers from 0-100 only using all of 1,2,3,4 and 5

Create all numbers from 0-100 only using 1,2,3,4 and 5. No repeats and you have to use each number. Also, you can use any operation. I've only gotten to 50 by pure brute force. I think that this might be fun for you puzzlers!

Also, could someone please help with tags and wording of this puzzle. I know this is similar Expressing numbers using 0, 1, 2, 3, and 4 but I'm still posting this.

Note: You may not use concatenation to combine digits. Ex 1 and 2 to get 21 or 12

• The question you linked is similar, but not identical; that one asked for the lowest number that couldn't be made from those five numbers. A couple of things, do the numbers have to stay in the order 1-2-3-4-5? Can we concatenate (for example, combining 1 and 2 into 12)? Commented Mar 20, 2017 at 14:55
• The original problem said that we can not use "glue" and combine digits or this would be really easy...
– user35295
Commented Mar 20, 2017 at 14:58
• Should I delete this? So sorry, I am new.
– user35295
Commented Mar 20, 2017 at 14:58
• don't because I already started on it :d. You can add that "no glue" restriction also. Commented Mar 20, 2017 at 14:59
• The question is awesome, and deserves to be like a new FizzBuzz test that includes recursion!
– smci
Commented Mar 22, 2017 at 9:02

Also, you can use any operation.

Ok then.

$\begin{array}{c|c} 0 & \log_{\frac1 2} \left( \log_{4!!-3} 5 \right) \\ 1 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt 5 \right) \\ 2 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt{\sqrt 5\,} \right) \\ 3 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt{\sqrt{\sqrt 5\,}\,} \right) \\ 4 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt{\sqrt{\sqrt{\sqrt 5\,}\,}\,} \right) \\ 5 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt{\sqrt{\sqrt{\sqrt{\sqrt 5\,}\,}\,}\,} \right) \\ 6 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt 5\,}\,}\,}\,}\,} \right) \\ 7 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt 5\,}\,}\,}\,}\,}\,} \right) \\ 8 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt 5\,}\,}\,}\,}\,}\,}\,} \right) \\ 9 & \log_{\frac1 2} \left( \log_{4!!-3} \sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{\sqrt 5\,}\,}\,}\,}\,}\,}\,}\,} \right) \\ \\ \vdots & \vdots \end{array}$

would you mind running me through this solution?

This is because for any number $x$ (and in my examples, $4!!-3=5$)

$$\log_\frac12\left[\log_x\underbrace{\sqrt{\sqrt{\dots\sqrt{x\,}\,}\,}}_\text{n square roots}\right]\equiv n$$

And the reason this is true

\begin{align} & a^b=c\implies \log_a c = b \\ & \sqrt x = x^{\frac12} \\ & \sqrt{\sqrt x} = x^{\frac14} \\ & \sqrt{\sqrt{\sqrt x}} = x^{\frac18} \\ \vdots \\ & \log_x \sqrt{\sqrt x} = \frac14 \\ & \log_\frac12 \left[ \log_x \sqrt{\sqrt x} \right] = \log_\frac12 \left(\frac14\right) = 2 \impliedby \left(\frac12\right)^2=\frac14 \end{align}

See what happens when you add another square root.

• Nice idea! Thats really cool!
– user35295
Commented Mar 21, 2017 at 1:24
• For the less math-ept - would you mind running me through this solution? Commented Mar 21, 2017 at 4:37
• What operation is !!? I had assumed factorial - factorial, but this makes for a very large number. Commented Mar 21, 2017 at 9:57
• @JamesWebster en.wikipedia.org/wiki/Double_factorial Commented Mar 21, 2017 at 10:59
• This is both brilliant and beautiful. Commented Mar 21, 2017 at 14:18

\begin{align} 0 & = (1 + 2 - 3) \times (4 + 5) \\ 1 & = 1 + (2+3-5) \times 4 \\ 2 & = 2 + (1+3-4) \times 5 \\ 3 & = 1 -2+3-4+5 \\ 4 & = 1 \times (2+3-5) + 4 \\ 5 & = 1-2-3+4+5 \\ 6 & = 5+1 \times (2+3-4) \\ 7 & = 5+2 \times (4-3) \times 1 \\ 8 & = 5+3 \times (4-2-1) \\ 9 & = 5+3 + (4-2-1) \\ 10 & = 5+4 + (3-2) \times 1 \\ 11 & = 5+4 + (3-2) + 1 \\ 12 & = 5+4+3 \times (2-1) \\ 13 & = 5+4+3 + 2-1 \\ 14 & = 5+4+3 + 2 \times 1 \\ 15 & = 5+4+3 + 2 + 1 \\ 16 & = 5+3 + 4 \times 2 \times 1 \\ 17 & = 5+3 + 4 \times 2 + 1 \\ 18 & = 5 + 4 + 3 ^ 2 \times 1 \\ 19 & = 5 + 4 + 3 ^ 2 + 1 \\ 20 & = 5 \times 4 \times (3 - 2) \times 1 \\ 21 & = 5 \times 4 \times (3 - 2) + 1 \\ 22 & = 5 \times 4 + 3 - 2 + 1 \\ 23 & = 5 \times 4 + 3 \times (2 - 1) \\ 24 & = 5 \times 4 + 3 + (2 - 1) \\ 25 & = 5^{4-3+2-1} \\ 26 & = (5 + 4 + 3 + 1) \times 2 \\ 27 & = 3 \times (4-1) \times (5-2) \\ 28 & = (5\times 3 - 1) \times \frac{4}{2} \\ 29 & = 5 \times 3 \times \frac{4}{2} - 1 \\ 30 & = 5 \times 3 \times \frac{4}{2} \times 1 \\ 31 & = 5 \times 3 \times \frac{4}{2} + 1 \\ 32 & = 2^5 + 4 -3 - 1 \\ 33 & = 2^5 + 4 - 3 \times 1 \\ 34 & = 2^5 + 4 - 3 + 1 \\ 35 & = (3 + 4) \times 5 \times (2-1) \\ 36 & = (3 + 4) \times 5 + (2-1) \\ 37 & = 2 \times 4 \times 5 - 3 \times 1 \\ 38 & = 2 \times 4 \times 5 - 3 + 1 \\ 39 & = (5 + 4 \times 2 ) \times 3 \times 1 \\ 40 & = (5 + 4 \times 2 ) \times 3 + 1 \\ 41 & = (4 + 5 \times 2 ) \times 3 - 1 \\ 42 & = (4 + 5 \times 2 ) \times 3 \times 1 \\ 43 & = (4 + 5 \times 2 ) \times 3 + 1 \\ 44 & = (1+2+3+5) \times 4 \\ 45 & = 3 \times 5 \times (4 - 2 +1) \\ 46 & = (5 \times 4 + 3 ) \times 2 \times 1 \\ 47 & = (5 \times 4 + 3 ) \times 2 + 1 \\ 48 & = 2^4 \times (5-3+1) \\ 49 & = (5+2) \times (4+3) \times 1 \\ 50 & = (5+2) \times (4+3) + 1 \\ 51 & = (5 \times 4 - 3) \times (2+1) \\ 52 & = (3 \times 5 - 2) \times 4 \times 1 \\ 53 & = (3 \times 5 - 2) \times 4 + 1 \\ 54 & = (5 \times 4 - 2) \times 3 \times 1 \\ 55 & = (5 \times 4 - 2) \times 3 + 1 \\ 56 & = (5+3 -1) \times 4^2 \\ 57 & = (5 \times 4 \times 3) -2 -1 \\ 58 & = (5 \times 4 \times 3) -2 \times 1 \\ 59 & = (5 \times 4 \times 3) -2 +1 \\ 60 & = (5 \times 4 \times 3) \times (2 -1) \\ 61 & = (5 \times 4 \times 3) + 2 - 1 \\ 62 & = (5 \times 4 \times 3) + 2 \times 1 \\ 63 & = (5 \times 4 \times 3) + 2 + 1 \\ 64 & = 2^5 \times (4-3) \times 1 \\ 65 & = 2^5 \times (4-3) + 1 \\ 66 & = 2^5 + 4-3+1 \\ 67 & = 4! \times 3 - 5 \times (2-1) \\ 68 & = 4! \times 3 - 5 + (2-1) \\ 69 & = (4+3) \times 5 \times 2 -1 \\ 70 & = (4+3) \times 5 \times 2 \times 1 \\ 71 & = (4+3) \times 5 \times 2 + 1 \\ 72 & = 4! \times (5-2) \times 1^3 \\ 73 & = 4! \times (5-2) + 1^3 \\ 74 & = 5^2 \times 3 - 1^4 \\ 75 & = 5^2 \times 3 \times 1^4 \\ 76 & = 5^2 \times 3 + 1^4 \\ 77 & = 4! \times 3 + 5 \times 1^2 \\ 78 & = 4! \times 3 + 5 + 1^2 \\ 79 & = 4! \times 3 + 5 + 2 \times 1 \\ 80 & = 4! \times 3 + 5 + 2 + 1 \\ 81 & = 3^{4-1} \times (5-2) \\ 82 & = 4! \times 3 + 5 \times 2 \times 1 \\ 83 & = 4! \times 3 + 5 \times 2 + 1 \\ 84 & = 3^4 + 5 - 2 \times 1 \\ 85 & = 3^4 + 5 - 2 + 1 \\ 86 & = 3^4 + 5 \times 1^2 \\ 87 & = 3^4 + 5 + 1^2 \\ 88 & = 3^4 + 5 +2^1 \\ 89 & = 3^4 + 5 + 2 + 1 \\ 90 & = 3^2 \times 5 \times 2^1 \\ 91 & = 3^2 \times 5 \times 2 + 1 \\ 92 & = 3^4 + 5 \times 2 + 1 \\ 93 & = 2^5 \times 3 +1 - 4 \\ 94 & = 5^2 \times 4 - 3! \times 1 \\ 95 & = 5^2 \times 4 - 3! + 1 \\ 96 & = 5^2 \times 4 - 3 - 1 \\ 97 & = 5^2 \times 4 - 3 \times 1 \\ 98 & = 5^2 \times 4 - 3 + 1 \\ 99 & = 5^2 \times 4 -1^3 \\ 100 & = 5^2 \times 4 \times 1^3 \end{align}

• You are quicker. Haha :) Commented Mar 20, 2017 at 15:05
• Just leave me a few more minutes. Commented Mar 20, 2017 at 15:05
• I'm done. Can someone please double check it. Commented Mar 20, 2017 at 16:03
• While trying to check whether we have any similarities, I noticed that you have missed the 3 in 72 and 73. You can fit it easily, though. :) Commented Mar 20, 2017 at 18:29
• Thanks Maria for the heads up and boboquack for the edit. That was my intent in the first place, but I was in a hurry and missed it. Commented Mar 20, 2017 at 21:01

$0 = 3^2 \times 1 - 4 - 5$
$1 = 3^2 + 1 - 4 - 5$
$2 = (4 - 5 + 3)/ 1^2$
$3 = (5 + 4) \times 1 - 2 \times 3$
$4 = (5 + 3) \times 1^2 - 4$
$5 = 5 \times 2 - 4 - 1^3$
$6 = 3 \times 2 + 5 - 1 - 4$
$7 = 5 \times 2 - 3 \times 1^4$
$8 = 4 + 5 - 3 + 2^1$
$9 = 3^2 - 4 - 1 + 5$
$10 = 5 \times 2 - 4 + 1 + 3$
$11 = 5 + 3 + 4 - 1^2$
$12 = 4 \times 5 \times 1 - 2^3$
$13 = 4 \times 5 - 1 - 3 \times 2$
$14 = 4 \times 5 - 3 \times 2 \times 1$
$15 = 1 + 2 + 3 + 4 + 5$
$16 = 4 + 5 + 2 \times 3 + 1$
$17 = 5 + 3 + 2 \times 4 + 1$
$18 = 5 + 4 + 1 + 2^3$
$19 = 5 + 4 + 1 + 3^2$
$20 = 5 \times 4 - 3 + 2 + 1$
$21 = 5 \times 4 + 3 - 2 \times 1$
$22 = 5 \times 4 + 3 - 2 + 1$
$23 = 4 \times 3 + 5 \times 2 + 1$
$24 = 5 \times 4 + 2 \times (3 - 1)$
$25 = 5 \times 4 + 3 + 2 \times 1$
$26 = 5 \times 4 + 3 + 2 + 1$
$27 = 5 \times 4 + 3 \times 2 + 1$
$28 = 5 \times 4 \times 1 + 2^3$
$29 = 5 \times 4 \times 1 + 3^2$
$30 = 5 \times 4 + 3^2 + 1$
$31 = 5 \times (4 + 2) + 1^3$
$32 = 5 \times (4+2) + 3 - 1$
$33 = 5 \times (4+2) + 3 \times 1$
$34 = 5 \times (4+2) + 3 + 1$
$35 = 5 \times 3 \times 2 + 4 + 1$
$36 = 5 \times (4+2) \times 1 + 3!$
$37 = 5 \times (4+2) + 3! + 1$
$38 = 4! + 2 \times 5 + 3 + 1$
$39 = 2^5 + 4 + 3 \times 1$
$40 = 2^5 + 4 + 3 + 1$
$41 = 5 \times 2^3 + 1^4$
$42 = 5 \times 3^2 + 4 - 1$
$43 = 5 \times 2^3 + 4 - 1$
$44 = 5 \times 2^3 + 4 \times 1$
$45 = 5 \times 2^3 + 4 + 1$
$46 = 5 \times 4 \times 2 \times 1 + 3!$
$47 = 5 \times 3(4 - 1) + 2$ or $5 \times 4 \times 2 + 1 + 3!$
$48 = 4! \times 2 \times 1^3$
$49 = 4! \times 2 + 1^{3+5}$
$50 = 4! \times 2 + 3 - 1^5$ or $5 \times (4+3+2+1)$
$51 = 4! \times 2 \times 1^5 + 3$
$52 = 4! \times 2 + 3 + 1^5$
$53 = 4! \times 2 + 5 \times 1^3$
$54 = 4! \times 2 + 5 + 1^3$
$55 = 4! \times 2 + 5 + 3 - 1$
$56 = 4! \times 2 + 5 + 3 \times 1$
$57 = 5 \times 4 \times 3 - 1 - 2$ or $4! \times 2 + 5 + 3 + 1$
$58 = 5 \times 4 \times 3 - 2 \times 1$
$59 = 5 \times 4 \times 3 - 2 + 1$
$60 = 5 \times 4 \times 3 \times 1^2$
$61 = 5 \times 4 \times 3 - 1 + 2$
$62 = 5 \times 4 \times 3 + 2 \times 1$
$63 = 5 \times 4 \times 3 + 1 + 2$
$64 = 2^5 \times (4 - 3 + 1)$
$65 = 4! \times 3 - 5 - 2 \times 1$
$66 = 4! \times 3 - 5 - 2 + 1$
$67 = 4! \times 3 - 5 \times 1^2$
$68 = 4! \times 3 - 5 - 1 + 2$
$69 = 4! \times 3 - 5 + 2 \times 1$
$70 = 4! \times 3 - 5 + 2 + 1$
$71 = 4! \times 3 - 1^{5+2}$
$72 = 4! \times 3 \times 1^{5+2}$
$73 = 4! \times 3 + 1^{5+2}$
$74 = 4! \times 3 + 2 \times 1^5$
$75 = 4! \times 3 \times 1 + 5 - 2$
$76 = 4! \times 3 + 5 - 2 + 1$
$77 = 4! \times 3 + 5 \times 1^2$
$78 = 4! \times 3 + 5 + 1^2$
$79 = 4! \times 3 + 5 + 2 \times 1$
$80 = 4! \times 3 + 5 + 2 + 1$
$81 = 4! \times 3 + 5 \times 2 - 1$
$82 = 4! \times 3 + 5 \times 2 \times 1$
$83 = 4! \times 3 + 5 \times 2 + 1$
$84 = 3^4 + 5 - 2 \times 1$
$85 = 3^4 + 5 - 1^2$
$86 = 3^4 + 5 \times 1^2$
$87 = 3^4 + 5 + 1^2$
$88 = 3^4 + 5 + 2 \times 1$
$89 = 3^4 + 5 + 2 + 1$
$90 = 3^4 + 5 \times 2 - 1$
$91 = 3^4 + 5 \times 2 \times 1$
$92 = 3^4 + 5 \times 2 + 1$
$93 = 4! \times (3 + 1) - 5 + 2$
$94 = 4! \times (5 - 1^3) - 2$
$95 = 4! \times (5 + 3)/2 - 1$
$96 = 4! \times 1 \times (5+3)/2$
$97 = 4! \times (5+3)/2 + 1$
$98 = 4! \times (5-1^3) + 2$
$99 = 4! \times (5-1^2) + 3$
$100 = 5 \times 4 \times (3+2) \times 1$
$101 = 5 \times 4 \times (3+2) + 1$

• I don't want to nitpick but it says numbers 0-100. You can add 0 at the top to make it legal. Commented Mar 20, 2017 at 16:04
• Oops copy pasting error - I did have it. Commented Mar 20, 2017 at 16:05
• Uhm, yes, I figured it was bugging it. I will reformat it when I get home later. For now, I would leave it like that - it is a little off, but visible. Commented Mar 20, 2017 at 16:07

I wrote a program in Python 3:

from collections import defaultdict
from functools import lru_cache
from itertools import combinations, chain, product
from operator import add, sub, mul, truediv, pow
import random

@lru_cache(maxsize=None)
def powerset(s):
return chain.from_iterable(combinations(s, r) for r in range(1, len(s)))

operations = [
('-', sub),
('*', mul),
('/', truediv),
('**', pow),
]

def bracket(s):
if len(s) > 1:
return '(' + s + ')'
return s

def sample(s):
return random.sample(s, min(len(s), 5))

@lru_cache(maxsize=None)
def calculate(nums):
if len(nums) == 1:
num = nums[0]
return {num: str(num)}
for left_parts in powerset(nums):
right_parts = tuple(x for x in nums if x not in left_parts)
for (left_num, left_strings), (right_num, right_strings) in product(
calculate(left_parts).items(),
calculate(right_parts).items()):
for op_str, op_func in operations:
orders = [list]
if op_str not in '+*':
orders += [reversed]
for order in orders:
try:
result_num = op_func(*order([left_num, right_num]))
except (ZeroDivisionError, OverflowError):
continue
if isinstance(result_num, complex) or abs(result_num) > 1000:
continue
for left_string, right_string in product(sample(left_strings),
sample(right_strings)):
result_string = '{1} {0} {2}'.format(op_str,
*order([bracket(left_string),
bracket(right_string)]))
assert eval(result_string) == result_num

def main():
answer = calculate((1, 2, 3, 4, 5))
nums = sorted(num for num in answer if int(num) == num and num >= 0)
for i, num in enumerate(nums):
if num != i:
break
print(int(num), '=',
.replace('**', '^')
.replace('*', '×'))

main()


It uses addition, subtraction, multiplication, division, exponentiation (x to the power of y), and parentheses. It gives up to 5 solutions for each number from 1 up to 177. It could give solutions for many more numbers but you'd start to see gaps, since apparently you can't make 178. Unfortunately it's not smart enough to tell when two solutions are essentially the same, so there's often lots of very similar looking solutions for a particular number, although I tried to mix them up. You can also get more solutions per number, but when I tried with 7 I hit the character limit in my answer.

0 = (2 + 3) - (5 ^ (1 ^ 4))  or  (3 ^ 2) - (4 + (5 × 1))  or  (((5 + 3) / 4) / 2) - 1  or  ((1 + 2) / 3) + (4 - 5)  or  (3 - (1 + 2)) ^ (4 / 5)
1 = 1 ^ (4 ^ (3 - (2 × 5)))  or  1 ^ (4 × (2 × (3 / 5)))  or  (1 - 2) ^ ((5 ^ 3) × 4)  or  1 ^ (5 + (4 + (2 - 3)))  or  (3 × 1) / (2 - (4 - 5))
2 = (1 + (3 - 2)) ^ (5 - 4)  or  2 - (5 ^ ((1 - 4) ^ 3))  or  ((1 ^ 5) ^ 3) × (4 - 2)  or  (3 - 1) / (4 + (2 - 5))  or  (3 - 4) - ((2 × 1) - 5)
3 = 3 ^ ((5 - 4) ^ (2 ^ 1))  or  (5 - 4) + ((1 + 3) / 2)  or  2 × (3 / ((5 - 4) + 1))  or  ((1 + 3) + 5) - (2 + 4)  or  (4 + 3) + ((2 - 5) - 1)
4 = (1 / 2) ^ (5 - (4 + 3))  or  (3 / 1) + (2 - (5 - 4))  or  (3 - (4 - 5)) / (2 - 1)  or  (4 + 3) - (2 + (1 ^ 5))  or  (2 - (5 - 3)) + (1 × 4)
5 = (4 + 5) - (3 - (1 - 2))  or  (3 - (1 + 5)) + (2 × 4)  or  (5 × 3) / ((4 + 1) - 2)  or  (5 + 3) - (1 - (2 - 4))  or  ((4 / 2) + 3) / (1 ^ 5)
6 = (4 × 3) + (1 - (5 + 2))  or  (3 × 4) + (1 - (5 + 2))  or  (4 × 3) - ((1 ^ 2) + 5)  or  (3 / (1 - 4)) + (2 + 5)  or  (2 - 5) × ((1 ^ 4) - 3)
7 = ((5 + 2) / (3 + 1)) × 4  or  (5 + 4) - (2 / (1 ^ 3))  or  (5 + (4 - 2)) × (1 ^ 3)  or  ((5 × 2) / (1 ^ 4)) - 3  or  (1 × 5) - (4 - (3 × 2))
8 = 3 + (5 ^ (4 - (2 + 1)))  or  ((2 × 3) - 4) × (5 - 1)  or  (1 × 5) + ((2 - 3) + 4)  or  2 + ((5 + (4 - 3)) × 1)  or  ((5 + 1) + 2) × (4 - 3)
9 = (3 ^ 2) - (4 - (5 - 1))  or  (4 × 1) - (5 / (2 - 3))  or  (5 + 2) - (4 / (1 - 3))  or  (1 ^ (3 / 5)) + (4 × 2)  or  3 - (((2 - 5) + 1) - 4)
10 = 2 + ((4 × (1 + 5)) / 3)  or  (5 × (1 ^ 3)) × (4 - 2)  or  (5 + (3 - 2)) + (4 ^ 1)  or  (5 - 4) + (3 ^ (2 / 1))  or  (3 + 1) / ((4 - 2) / 5)
11 = (4 - (2 - 1)) + (5 + 3)  or  ((3 × 5) - 4) × (1 ^ 2)  or  (3 ^ 2) - ((4 - 5) - 1)  or  (5 + 4) - ((2 - 1) - 3)  or  (3 + (2 × 4)) ^ (1 ^ 5)
12 = (1 × 2) × (5 + (4 - 3))  or  (5 × 4) - ((1 × 2) ^ 3)  or  ((5 + 1) × 2) × (4 - 3)  or  ((1 + 5) × (2 ^ 3)) / 4  or  (3 × ((5 - 1) ^ 2)) / 4
13 = (3 + ((4 + 2) - 1)) + 5  or  (3 + 1) × (2 + (5 / 4))  or  (2 + (4 + 3)) - (1 - 5)  or  (5 + (2 + 3)) + (4 - 1)  or  ((5 - (1 + 2)) ^ 4) - 3
14 = (5 - (2 - (3 × 4))) - 1  or  (2 ^ 4) + ((1 ^ 5) - 3)  or  (3 - 1) × ((4 + 5) - 2)  or  ((4 + (3 + 5)) + 2) / 1  or  (2 × (4 + 3)) ^ (1 ^ 5)
15 = (3 × 2) + ((4 + 5) ^ 1)  or  (1 × (5 + 4)) + (3 × 2)  or  ((4 + 5) × 2) - (3 ^ 1)  or  1 - ((5 - (4 ^ 2)) - 3)  or  (2 ^ 4) + (3 - (5 - 1))
16 = ((5 × 3) + 4) - (1 + 2)  or  ((4 × 3) + 1) + (5 - 2)  or  (3 + 5) / (2 × (1 / 4))  or  (5 + (4 × 2)) + (1 × 3)  or  (4 + (5 - 1)) + (2 ^ 3)
17 = 1 - (((3 - 5) × 4) × 2)  or  (3 ^ 4) - (2 ^ (5 + 1))  or  (5 × 3) - ((2 - 4) / 1)  or  2 - (5 × ((1 ^ 3) - 4))  or  ((2 + 5) × 3) - (4 / 1)
18 = (4 × 3) × ((5 / 2) - 1)  or  ((5 × 3) + 1) + (4 - 2)  or  (5 × 4) - ((1 + 3) - 2)  or  (5 × 3) + (1 - (2 - 4))  or  (2 + 4) + ((5 - 1) × 3)
19 = ((5 ^ (3 - 1)) - 2) - 4  or  (2 × 5) - (3 × (1 - 4))  or  (2 - 3) + ((1 × 5) × 4)  or  ((4 × 3) + 2) + (5 ^ 1)  or  (4 / (2 - 1)) + (5 × 3)
20 = (1 + 5) × (2 + (4 / 3))  or  ((5 × 4) + 3) - (2 + 1)  or  (4 × 2) / (1 - (3 / 5))  or  (2 ^ 4) / ((3 + 1) / 5)  or  (3 - 1) + (2 × (5 + 4))
21 = 3 - ((2 / 1) - (4 × 5))  or  ((5 × 1) - 2) × (4 + 3)  or  (3 ^ 1) × (5 - (2 - 4))  or  (5 × 4) - ((1 ^ 3) - 2)  or  ((5 × 4) + 3) - (1 × 2)
22 = ((5 × (2 + 1)) + 3) + 4  or  (2 + 5) + (3 × (4 + 1))  or  ((5 × 3) - (4 / 1)) × 2  or  (3 + (5 + (4 - 1))) × 2  or  ((3 × 1) × 4) + (2 × 5)
23 = (4 ^ 2) + (3 + (5 - 1))  or  ((2 × 4) + (5 × 3)) / 1  or  (2 × 5) + ((4 × 3) + 1)  or  (3 - 1) - (4 - (5 ^ 2))  or  1 - (3 - (5 ^ (4 / 2)))
24 = ((2 ^ 4) + (3 × 1)) + 5  or  (2 ^ 1) × ((5 + 4) + 3)  or  (5 ^ 2) - (4 - (3 ^ 1))  or  (3 × 1) + ((4 ^ 2) + 5)  or  ((2 × 4) × 3) / (1 ^ 5)
25 = ((2 + 3) × 5) × (1 ^ 4)  or  (3 + (5 × 4)) + (2 × 1)  or  ((3 × (2 ^ 5)) / 4) + 1  or  ((2 × 5) ^ (3 - 1)) / 4  or  5 + ((3 + (2 ^ 4)) + 1)
26 = (1 × 4) + ((5 ^ 2) - 3)  or  (4 + 1) + (3 × (5 + 2))  or  (2 × 3) + (4 × (5 × 1))  or  (3 ^ (1 + 2)) + (4 - 5)  or  ((2 × 3) × (5 / 1)) - 4
27 = 1 / (3 ^ ((4 - 5) - 2))  or  (3 × 1) ^ ((4 × 2) - 5)  or  (2 × (3 ^ 4)) / (5 + 1)  or  ((5 - 2) ^ 4) / (3 ^ 1)  or  (2 - (4 - 5)) ^ (1 × 3)
28 = (4 × (5 + 2)) × (1 ^ 3)  or  (5 + 2) × ((1 ^ 3) × 4)  or  ((4 × 5) ^ 1) + (2 ^ 3)  or  (5 × (3 + 2)) - (1 - 4)  or  (4 + (5 ^ 2)) - (1 ^ 3)
29 = (2 + 3) + (4 × (1 + 5))  or  (2 × (4 × 3)) + (1 × 5)  or  ((4 / 2) ^ 5) - (3 ^ 1)  or  (1 × 5) + ((2 × 3) × 4)  or  (1 ^ 4) + (3 + (5 ^ 2))
30 = ((1 + 4) × (3 + 2)) + 5  or  (4 × 3) × ((5 ^ 1) / 2)  or  ((4 / 2) + 3) × (1 + 5)  or  ((5 + 2) + 3) × (4 - 1)  or  (1 - ((2 - 4) ^ 5)) - 3
31 = (2 × 3) + ((1 + 4) × 5)  or  (4 ^ (5 / 2)) - (1 ^ 3)  or  1 × (4 - ((2 - 5) ^ 3))  or  ((4 × (2 + 1)) × 3) - 5  or  ((4 + 1) × 5) + (3 × 2)
32 = (4 × (5 + 3)) ^ (2 - 1)  or  (2 ^ 5) × (4 - (3 / 1))  or  (2 × 4) × (3 + (1 ^ 5))  or  (2 / 1) × ((3 - 5) ^ 4)  or  2 / (1 × (4 ^ (3 - 5)))
33 = 1 - ((2 / (3 - 4)) ^ 5)  or  (4 / 1) + ((2 ^ 5) - 3)  or  (2 ^ 5) - (3 - (1 × 4))  or  (4 + (5 + 2)) × (3 × 1)  or  ((2 ^ 5) - (3 - 4)) / 1
34 = (3 - 1) - ((2 - 4) ^ 5)  or  ((1 - 3) + 4) + (2 ^ 5)  or  ((4 / 2) ^ 5) + (3 - 1)  or  ((4 + 1) × 5) + (3 ^ 2)  or  2 - (((3 - 4) - 1) ^ 5)
35 = (2 × ((3 × 4) + 5)) + 1  or  (1 × 3) - ((2 - 4) ^ 5)  or  5 / ((1 ^ 2) / (4 + 3))  or  (1 + ((4 ^ 3) + 5)) / 2  or  ((4 / 2) ^ 5) + (1 × 3)
36 = 4 × ((3 + 5) + (1 ^ 2))  or  (5 × 4) + ((3 + 1) ^ 2)  or  ((5 + 4) + 3) × (2 + 1)  or  (4 + 2) × ((1 ^ 3) + 5)  or  (3 × (5 - 2)) × (1 × 4)
37 = ((3 + 1) × (2 × 4)) + 5  or  ((5 × 2) × 4) - (3 × 1)  or  ((2 × 5) × 4) - (3 ^ 1)  or  (((2 × 5) ^ 1) × 4) - 3  or  (5 / (1 / (4 + 3))) + 2
38 = (3 × (5 + (4 × 2))) - 1  or  (2 ^ 1) × (4 + (5 × 3))  or  (2 / 1) × (4 + (3 × 5))  or  ((4 + (3 × 5)) × 2) × 1  or  ((3 - 1) ^ 5) + (2 + 4)
39 = (1 + (3 × 4)) × (5 - 2)  or  (5 × (2 × 4)) - (1 ^ 3)  or  3 / (1 / (5 + (2 × 4)))  or  (((2 ^ 5) + 3) + 4) ^ 1  or  1 - ((5 - (3 ^ 4)) / 2)
40 = (2 / ((1 ^ 3) / 4)) × 5  or  ((5 / 1) + (2 + 3)) × 4  or  5 / ((2 / (4 × 1)) ^ 3)  or  (3 + 2) × ((4 - 1) + 5)  or  ((3 + 5) + 2) × (1 × 4)
41 = ((5 ^ 3) - 2) / (4 - 1)  or  1 × (((3 ^ 2) × 4) + 5)  or  (4 × (3 ^ 2)) + (5 ^ 1)  or  (1 ^ 3) + (5 × (4 × 2))  or  (2 × ((4 × 5) - 1)) + 3
42 = (3 × ((5 × 2) + 4)) × 1  or  (2 + 4) × (5 - (1 - 3))  or  2 × ((5 ^ (3 - 1)) - 4)  or  (2 + 5) × ((4 + 3) - 1)  or  1 + ((4 × (3 ^ 2)) + 5)
43 = (((2 ^ 4) × 3) / 1) - 5  or  (3 + ((5 × 2) × 4)) ^ 1  or  ((3 × (4 ^ 2)) - 5) × 1  or  1 / (2 / (5 + (3 ^ 4)))  or  (3 × 1) + ((5 × 2) × 4)
44 = (2 ^ 5) + (3 × (4 / 1))  or  (4 ^ 1) × (5 + (2 × 3))  or  (3 + ((5 × 4) × 2)) + 1  or  4 / (1 / (5 + (2 × 3)))  or  (2 ^ 5) + ((3 / 1) × 4)
45 = (4 + 5) × ((2 / 1) + 3)  or  (4 + 5) / (1 / (2 + 3))  or  (3 × 4) + (1 + (2 ^ 5))  or  (1 + (4 - 2)) × (5 × 3)  or  (3 + 2) × (4 + (1 × 5))
46 = (2 / 1) × (3 + (4 × 5))  or  (3 × (1 + (4 ^ 2))) - 5  or  ((2 + 3) × (5 + 4)) + 1  or  (2 × (3 + (4 × 5))) × 1  or  (2 / 1) × (3 + (5 × 4))
47 = (5 × 4) + (3 ^ (2 + 1))  or  3 + ((1 + (2 × 5)) × 4)  or  2 + (((4 - 1) × 3) × 5)  or  (2 × ((5 × 4) + 3)) + 1  or  ((1 + 4) × 3) + (2 ^ 5)
48 = (3 + 1) × (4 × (5 - 2))  or  ((2 + 4) × (3 + 5)) × 1  or  ((5 + 4) - 1) × (2 × 3)  or  ((5 - 3) ^ 4) × (1 + 2)  or  (4 + 2) × ((3 + 5) ^ 1)
49 = ((2 + 5) × 1) × (3 + 4)  or  ((5 - (3 × 4)) ^ 2) / 1  or  (5 - (4 × 3)) ^ (2 / 1)  or  (3 + 4) ^ (2 × (1 ^ 5))  or  (1 - (5 + 3)) ^ (4 - 2)
50 = 5 × ((4 + (1 + 3)) + 2)  or  ((3 - (1 / 2)) × 5) × 4  or  ((4 × 3) - 2) / (1 / 5)  or  ((4 + 1) ^ 3) × (2 / 5)  or  (5 × 1) × (4 + (2 × 3))
51 = ((4 ^ 2) + (1 ^ 5)) × 3  or  (2 + 1) × ((4 × 3) + 5)  or  ((3 × 4) + 5) × (1 + 2)  or  3 + (4 × ((5 + 1) × 2))  or  (5 × (4 + (3 × 2))) + 1
52 = (5 - 1) + (3 × (4 ^ 2))  or  (4 × 1) × (3 + (2 × 5))  or  (5 + ((2 ^ 4) × 3)) - 1  or  (5 + (4 × 2)) × (1 + 3)  or  (3 ^ (4 - 1)) + (5 ^ 2)
53 = 1 + (4 × (3 + (5 × 2)))  or  (5 + ((2 ^ 4) × 3)) ^ 1  or  (4 × (3 + (5 × 2))) + 1  or  5 + (1 × ((4 ^ 2) × 3))  or  (5 / 1) + ((4 ^ 2) × 3)
54 = (5 + 1) + (3 × (2 ^ 4))  or  (3 × 1) × ((5 + 4) × 2)  or  (1 + 5) × ((3 + 2) + 4)  or  (3 × (5 - (1 / 2))) × 4  or  ((4 + 5) / 1) × (2 × 3)
55 = (4 + 1) × ((3 × 2) + 5)  or  (1 × 5) × (3 + (4 × 2))  or  (5 × (3 + (2 × 4))) / 1  or  ((4 ^ 3) + 1) - (2 × 5)  or  ((3 × 2) + 5) × (1 + 4)
56 = (3 ^ 4) - (5 ^ (2 / 1))  or  ((3 ^ 1) ^ 4) - (5 ^ 2)  or  (2 × (3 + 4)) × (5 - 1)  or  (2 + 5) × ((1 + 4) + 3)  or  (3 + 5) × ((2 + 4) + 1)
57 = ((4 ^ 3) - (5 + 2)) ^ 1  or  ((2 ^ (1 + 5)) - 4) - 3  or  ((5 × (2 + 1)) × 4) - 3  or  (4 ^ (1 × 3)) - (2 + 5)  or  (((4 × 5) - 2) + 1) × 3
58 = 4 + ((3 ^ 2) × (5 + 1))  or  ((5 + 1) × (3 ^ 2)) + 4  or  (1 + ((4 ^ 3) - 5)) - 2  or  (((3 × 5) - 1) × 4) + 2  or  (3 × (5 × 4)) - (2 × 1)
59 = (((3 + 5) ^ 2) - 1) - 4  or  (4 ^ 3) - (5 ^ (2 - 1))  or  (4 ^ 3) - (5 / (1 ^ 2))  or  ((5 × (4 × 3)) - 2) + 1  or  (1 ^ 2) × ((4 ^ 3) - 5)
60 = (5 × (3 ^ (1 ^ 2))) × 4  or  (5 ^ 1) × ((2 ^ 3) + 4)  or  (3 × (4 × 5)) × (1 ^ 2)  or  ((1 ^ 2) × 4) × (3 × 5)  or  ((4 ^ 3) - 1) - (5 - 2)
61 = ((2 - 5) / 1) + (4 ^ 3)  or  (4 ^ (5 - 2)) - (3 ^ 1)  or  (4 ^ (3 × 1)) + (2 - 5)  or  (2 / 1) - (5 - (4 ^ 3))  or  (1 × 2) + ((4 ^ 3) - 5)
62 = ((4 ^ (1 ^ 5)) ^ 3) - 2  or  1 + ((4 ^ 3) + (2 - 5))  or  1 - ((5 - (4 ^ 3)) - 2)  or  1 × (2 + ((5 × 4) × 3))  or  ((4 ^ (2 + 1)) - 5) + 3
63 = (3 × 1) × ((5 ^ 2) - 4)  or  3 - (4 - (2 ^ (5 + 1)))  or  (3 × (5 + (2 ^ 4))) / 1  or  (5 × (4 × 3)) + (1 + 2)  or  (3 × (5 × 4)) + (2 + 1)
64 = (2 ^ (4 + 1)) × (5 - 3)  or  (2 × 4) / (1 / (5 + 3))  or  (3 + (1 ^ 5)) × (4 ^ 2)  or  (1 ^ 3) × (4 ^ (5 - 2))  or  (4 × 2) ^ ((5 - 3) / 1)
65 = ((3 ^ 2) + 4) × (5 ^ 1)  or  (5 ^ 1) × ((2 ^ 4) - 3)  or  ((5 × 2) + 3) × (4 + 1)  or  4 + ((2 ^ (1 + 5)) - 3)  or  (3 × (2 + (4 × 5))) - 1
66 = 4 × ((5 + (1 / 2)) × 3)  or  4 - (2 + ((1 - 5) ^ 3))  or  2 × ((4 × (5 + 3)) + 1)  or  ((4 / (1 ^ 5)) ^ 3) + 2  or  (4 / 2) + ((5 - 1) ^ 3)
67 = ((4 ^ 3) + 5) - (1 × 2)  or  ((1 ^ 5) + (4 ^ 3)) + 2  or  1 + ((2 + (5 × 4)) × 3)  or  (5 - 2) + (4 ^ (3 ^ 1))  or  (4 ^ 3) + ((5 - 2) / 1)
68 = 5 + ((4 ^ 3) - (1 ^ 2))  or  1 - ((2 - 5) - (4 ^ 3))  or  ((5 × 3) + 2) × (4 ^ 1)  or  ((3 × 5) + 2) × (4 × 1)  or  ((3 × 5) + 2) × (4 / 1)
69 = (4 ^ 3) - (5 × (1 - 2))  or  1 + (((5 × 3) + 2) × 4)  or  (1 + 2) × ((4 × 5) + 3)  or  ((4 ^ 3) + 5) × (1 ^ 2)  or  (((4 + 3) × 5) × 2) - 1
70 = (2 / 1) × ((3 + 4) × 5)  or  ((3 ^ 4) - (5 × 2)) - 1  or  (5 × ((3 + 4) × 2)) ^ 1  or  5 × ((2 / 1) + (4 × 3))  or  (4 + 3) × (1 × (2 × 5))
71 = (5 + (4 ^ 3)) + (2 × 1)  or  (3 ^ 4) - (5 / (1 / 2))  or  (3 ^ 4) - (5 × (2 / 1))  or  (3 ^ 4) - (2 / (1 / 5))  or  (3 ^ (4 / 1)) - (2 × 5)
72 = (5 + (2 - 1)) × (3 × 4)  or  (4 + (2 ^ 3)) × (1 + 5)  or  (2 ^ 3) × (4 + (1 × 5))  or  (2 + 5) + ((4 ^ 3) + 1)  or  (3 × (5 ^ 2)) + (1 - 4)
73 = ((3 ^ 4) - 1) - (2 + 5)  or  ((3 ^ 4) - (1 + 2)) - 5  or  ((3 × 5) × (4 + 1)) - 2  or  ((3 ^ 4) - (2 + 1)) - 5  or  ((5 + 4) × (2 ^ 3)) + 1
74 = ((5 × 2) + (4 ^ 3)) ^ 1  or  (((3 ^ 4) - 5) - 2) ^ 1  or  (5 × 2) + ((4 ^ 1) ^ 3)  or  1 × ((5 × 2) + (4 ^ 3))  or  (3 ^ (4 ^ 1)) - (5 + 2)
75 = 3 × ((1 × 5) ^ (4 - 2))  or  (1 + ((3 ^ 4) - 2)) - 5  or  ((4 - 1) + 2) × (5 × 3)  or  (5 × (4 - 1)) × (3 + 2)  or  (1 + (3 ^ 4)) - (5 + 2)
76 = ((2 ^ 4) + 3) × (5 - 1)  or  (3 / (5 ^ (2 - 4))) + 1  or  ((5 × (2 ^ 4)) - 1) - 3  or  ((2 ^ 4) × 5) - (3 + 1)  or  (3 × (5 ^ 2)) + (1 ^ 4)
77 = (5 × (4 ^ 2)) - (3 / 1)  or  (((2 ^ 4) × 5) × 1) - 3  or  ((3 ^ 4) - 1) + (2 - 5)  or  ((4 ^ 2) / (1 / 5)) - 3  or  ((5 × 2) + 1) × (3 + 4)
78 = 3 × ((5 ^ (4 - 2)) + 1)  or  (4 × (5 × (3 + 1))) - 2  or  (((5 + 4) ^ 2) - 3) ^ 1  or  ((5 × (4 ^ 2)) - 3) + 1  or  (((2 - 5) ^ 1) ^ 4) - 3
79 = (2 + (1 - 5)) + (3 ^ 4)  or  (1 + ((5 + 4) ^ 2)) - 3  or  (3 ^ 4) - ((1 ^ 5) × 2)  or  ((3 ^ 4) - 2) ^ (1 ^ 5)  or  ((1 + 2) ^ 4) - (5 - 3)
80 = (5 × (2 × (3 - 1))) × 4  or  ((2 - (1 - 3)) × 5) × 4  or  (2 ^ 3) × ((4 + 1) + 5)  or  (3 ^ 4) - (1 ^ (2 ^ 5))  or  (((4 + 1) ^ 2) × 3) + 5
81 = (4 + 5) ^ ((3 - 2) + 1)  or  ((4 ^ 2) × 5) + (1 ^ 3)  or  ((3 + 1) + 5) ^ (4 / 2)  or  ((1 + (4 / 2)) ^ 5) / 3  or  ((2 ^ 3) + 1) × (4 + 5)
82 = (5 × (1 + (4 ^ 2))) - 3  or  (5 × ((3 - 1) ^ 4)) + 2  or  (3 × ((5 ^ 2) + 1)) + 4  or  (1 ^ 3) + ((2 - 5) ^ 4)  or  (((2 ^ 4) + 1) × 5) - 3
83 = 5 + ((3 ^ 4) - (2 + 1))  or  (3 + (5 × (4 ^ 2))) ^ 1  or  ((3 ^ 4) + (5 - 1)) - 2  or  (5 + (3 ^ 4)) - (1 + 2)  or  (3 + ((4 ^ 2) × 5)) × 1
84 = ((4 × (5 + 2)) ^ 1) × 3  or  (4 × (5 + 2)) × (3 / 1)  or  (4 ^ 1) × (3 × (5 + 2))  or  (5 + 2) / ((1 / 3) / 4)  or  1 + (3 + (5 × (2 ^ 4)))
85 = (3 + 1) + ((5 + 4) ^ 2)  or  ((5 - 2) ^ 4) + (1 + 3)  or  (1 + ((3 ^ 4) - 2)) + 5  or  (3 + ((5 + 4) ^ 2)) + 1  or  ((5 - 2) ^ 4) + (3 + 1)
86 = (3 ^ 4) - (5 / (1 - 2))  or  (((5 ^ 2) + 4) × 3) - 1  or  (((5 - 1) ^ 4) + 2) / 3  or  5 - ((1 - 2) × (3 ^ 4))  or  (((1 - 5) ^ 4) + 2) / 3
87 = 2 - ((1 - (3 ^ 4)) - 5)  or  2 + (5 + ((3 ^ 4) - 1))  or  2 - ((1 - 5) - (3 ^ 4))  or  ((3 ^ 4) + 2) - (1 - 5)  or  (3 ^ 4) + (5 + (1 ^ 2))
88 = ((5 ^ 2) - 3) × (4 ^ 1)  or  (2 + (5 + (3 ^ 4))) ^ 1  or  (2 + (3 ^ 4)) + (1 × 5)  or  ((1 + (2 ^ 4)) × 5) + 3  or  ((5 + (3 ^ 4)) + 2) × 1
89 = (2 + (3 ^ 4)) + (1 + 5)  or  (3 ^ (5 - 1)) + (4 × 2)  or  (4 ^ (3 ^ 1)) + (5 ^ 2)  or  1 - ((3 - (5 ^ 2)) × 4)  or  (((1 + 2) ^ 4) + 5) + 3
90 = (1 + (3 ^ 2)) × (4 + 5)  or  ((4 + 2) × 1) × (5 × 3)  or  2 × (3 × ((4 - 1) × 5))  or  ((4 + 2) × 5) × (3 / 1)  or  (3 × (4 + 2)) × (5 / 1)
91 = 1 + ((3 × 5) × (4 + 2))  or  ((3 × (2 ^ 5)) - 1) - 4  or  (3 ^ (1 × 4)) + (5 × 2)  or  (((2 ^ 5) × 3) - 4) - 1  or  (5 × 2) + (3 ^ (4 × 1))
92 = (1 + (3 ^ 4)) + (5 × 2)  or  (3 ^ 4) + (1 + (2 × 5))  or  (((5 ^ 2) + 1) - 3) × 4  or  (((5 ^ 2) - 3) + 1) × 4  or  1 × ((3 × (2 ^ 5)) - 4)
93 = (((4 - 2) ^ 5) - 1) × 3  or  (1 + (5 × (4 + 2))) × 3  or  ((1 + 5) × 2) + (3 ^ 4)  or  3 × (1 + (5 × (2 + 4)))  or  (((4 / 2) ^ 5) - 1) × 3
94 = (5 × ((2 ^ 4) + 3)) - 1  or  (5 × (3 + (4 ^ 2))) - 1  or  (((4 ^ 2) + 3) × 5) - 1  or  (5 × ((4 ^ 2) + 3)) - 1  or  (((2 ^ 4) + 3) × 5) - 1
95 = ((4 × 5) - 1) × (3 + 2)  or  ((2 ^ 5) + (4 ^ 3)) - 1  or  ((2 ^ 4) + 3) × (5 / 1)  or  ((3 + (4 ^ 2)) × 5) ^ 1  or  (1 × 5) × ((4 ^ 2) + 3)
96 = (5 - 1) × (3 × (2 × 4))  or  (2 ^ 5) × ((1 ^ 4) × 3)  or  ((2 + 1) + 5) × (3 × 4)  or  ((5 × 3) + 1) × (2 + 4)  or  ((4 × 2) × 3) × (5 - 1)
97 = (((5 ^ 2) × 4) - 3) × 1  or  ((5 ^ 2) × 4) - (3 × 1)  or  (2 ^ 4) + (3 ^ (5 - 1))  or  ((5 ^ 2) × (4 / 1)) - 3  or  ((5 ^ 2) × 4) - (3 ^ 1)
98 = (4 × (5 ^ 2)) + (1 - 3)  or  ((4 × (5 ^ 2)) - 3) + 1  or  1 - (3 - ((5 ^ 2) × 4))  or  (4 × (5 ^ 2)) - (3 - 1)  or  (((5 ^ 2) × 4) + 1) - 3
99 = ((2 ^ 4) × (5 + 1)) + 3  or  ((1 + 5) × (2 ^ 4)) + 3  or  (4 + ((2 ^ 5) × 3)) - 1  or  3 × ((1 ^ 4) + (2 ^ 5))  or  ((3 + 2) × (5 × 4)) - 1
100 = (((2 + 3) × 5) × 4) ^ 1  or  ((5 × 4) ^ 2) / (3 + 1)  or  (2 + 3) × (5 / (1 / 4))  or  ((3 + 2) × 4) × (1 × 5)  or  (4 / 1) × (5 × (2 + 3))
101 = ((5 ^ 2) × 4) + (1 ^ 3)  or  (1 + 4) + (3 × (2 ^ 5))  or  (((5 ^ 2) + 1) × 4) - 3  or  1 + (((2 ^ 5) × 3) + 4)  or  ((2 ^ 5) × 3) + (1 + 4)
102 = (3 + ((5 ^ 2) × 4)) - 1  or  3 + (((5 ^ 2) × 4) - 1)  or  2 + (4 × (5 ^ (3 - 1)))  or  3 - (1 - ((5 ^ 2) × 4))  or  (3 - 1) + (4 × (5 ^ 2))
103 = ((5 ^ 2) × 4) + (3 / 1)  or  (((5 ^ 2) × 4) / 1) + 3  or  3 + (4 × ((5 ^ 2) × 1))  or  (1 × 3) + (4 × (5 ^ 2))  or  (4 × (5 ^ 2)) + (3 / 1)
104 = 1 + (3 + ((5 ^ 2) × 4))  or  (3 + (1 / 4)) × (2 ^ 5)  or  (((2 + 3) × 5) + 1) × 4  or  (((3 - 1) × 5) ^ 2) + 4  or  ((4 × (5 ^ 2)) + 3) + 1
105 = (4 + 3) × ((1 + 2) × 5)  or  (5 ^ 2) - (1 - (3 ^ 4))  or  (5 + 2) × ((1 + 4) × 3)  or  (3 × (5 + 2)) × (4 + 1)  or  (1 + 4) × ((2 + 5) × 3)
106 = ((3 × 1) ^ 4) + (5 ^ 2)  or  (3 ^ 4) + ((5 / 1) ^ 2)  or  (5 ^ 2) + (3 ^ (4 / 1))  or  ((5 ^ 2) / 1) + (3 ^ 4)  or  ((3 ^ 4) + (5 ^ 2)) / 1
107 = (((2 ^ 5) + 4) × 3) - 1  or  3 + ((1 + (5 ^ 2)) × 4)  or  (((5 ^ 2) + 1) × 4) + 3  or  (3 ^ 4) + ((5 ^ 2) + 1)  or  1 + ((5 ^ 2) + (3 ^ 4))
108 = (4 + 2) × (3 × (5 + 1))  or  ((5 - 2) ^ 3) × (4 × 1)  or  (3 × 4) × ((5 × 2) - 1)  or  ((5 + 1) × 3) × (4 + 2)  or  2 × (((5 + 1) ^ 3) / 4)
109 = (5 ^ 3) - (4 ^ (2 ^ 1))  or  ((5 ^ 3) - (4 ^ 2)) ^ 1  or  1 + (4 × (3 ^ (5 - 2)))  or  ((1 × 5) ^ 3) - (2 ^ 4)  or  (5 ^ 3) - ((2 ^ 4) / 1)
110 = 5 × (((4 + 1) ^ 2) - 3)  or  (2 × 5) × ((3 × 4) - 1)  or  (((4 + 1) ^ 2) - 3) × 5  or  (5 ^ 3) + (1 - (2 ^ 4))  or  (((3 × 4) - 1) × 5) × 2
111 = ((3 + (5 ^ 2)) × 4) - 1  or  (4 + (1 + (2 ^ 5))) × 3  or  ((4 + 1) + (2 ^ 5)) × 3  or  3 × (((2 ^ 5) + 4) + 1)  or  3 × (((2 ^ 5) + 1) + 4)
112 = (3 + (5 ^ 2)) × (4 ^ 1)  or  (4 × (3 + (5 ^ 2))) ^ 1  or  (3 ^ 4) + ((2 ^ 5) - 1)  or  2 × (4 × ((3 × 5) - 1))  or  (4 × (2 + 5)) × (1 + 3)
113 = (4 × ((2 + 1) ^ 3)) + 5  or  ((3 ^ 4) / 1) + (2 ^ 5)  or  ((1 × 2) ^ 5) + (3 ^ 4)  or  ((2 ^ 5) + (3 ^ 4)) ^ 1  or  ((2 ^ 5) × 1) + (3 ^ 4)
114 = ((4 ^ 2) + 3) × (5 + 1)  or  (2 ^ 5) + (1 + (3 ^ 4))  or  (1 + (2 ^ 5)) + (3 ^ 4)  or  (3 + (2 ^ 4)) × (1 + 5)  or  1 + ((3 ^ 4) + (2 ^ 5))
115 = 5 × (((1 + 2) ^ 3) - 4)  or  5 × ((2 × (3 × 4)) - 1)  or  (4 × ((2 ^ 5) - 3)) - 1  or  (((3 × 2) × 4) - 1) × 5  or  5 × (((2 × 4) × 3) - 1)
116 = (3 + ((5 ^ 2) + 1)) × 4  or  4 × (1 × ((2 ^ 5) - 3))  or  (4 × 1) × ((2 ^ 5) - 3)  or  (((3 × 4) - 1) ^ 2) - 5  or  2 × ((4 ^ 3) - (1 + 5))
117 = ((5 ^ 3) ^ 1) - (2 × 4)  or  ((1 × 5) ^ 3) - (2 × 4)  or  (5 ^ 3) - (4 × (2 × 1))  or  1 × ((5 ^ 3) - (4 × 2))  or  ((5 ^ 3) × 1) - (2 × 4)
118 = (((3 ^ 5) + 1) / 2) - 4  or  ((5 ^ 3) - (2 + 4)) - 1  or  1 + ((5 ^ 3) - (2 × 4))  or  ((5 ^ 3) - (4 + 2)) - 1  or  (((5 ^ 3) - 4) - 2) - 1
119 = (5 ^ (4 - 1)) - (2 × 3)  or  ((5 ^ 3) - (4 + 2)) / 1  or  ((5 ^ 3) - (4 × 1)) - 2  or  ((5 ^ 3) - (4 / 1)) - 2  or  ((5 ^ 3) - 2) - (1 × 4)
120 = ((2 + (4 - 1)) ^ 3) - 5  or  2 / (((1 / 3) / 5) / 4)  or  (1 × 4) × (2 × (5 × 3))  or  (1 × (4 × 5)) × (3 × 2)  or  (2 × 3) × ((4 ^ 1) × 5)
121 = (4 × (1 - 2)) + (5 ^ 3)  or  (4 - (3 × 5)) ^ (2 / 1)  or  ((5 × 3) - 4) ^ (2 ^ 1)  or  (5 - (2 ^ 4)) ^ (3 - 1)  or  ((2 + 4) + 5) ^ (3 - 1)
122 = (2 × (4 ^ 3)) - (1 + 5)  or  ((5 ^ 3) + 2) - (4 + 1)  or  (2 × ((3 ^ 5) + 1)) / 4  or  ((3 ^ 5) + 1) / (4 - 2)  or  ((5 ^ 3) - (4 + 1)) + 2
123 = ((2 ^ (3 + 4)) - 5) ^ 1  or  1 × (((5 ^ 3) + 2) - 4)  or  ((5 ^ 3) - 4) + (2 ^ 1)  or  (((5 ^ 3) + 2) × 1) - 4  or  (2 / 1) - (4 - (5 ^ 3))
124 = (2 + 1) + ((5 ^ 3) - 4)  or  (2 + (5 ^ 3)) + (1 - 4)  or  3 + ((5 ^ (2 + 1)) - 4)  or  2 + ((5 ^ 3) + (1 - 4))  or  ((5 ^ 3) + 2) - (4 - 1)
125 = (5 × (3 + 2)) × (1 + 4)  or  ((5 ^ 4) / 1) / (3 + 2)  or  ((3 + 2) ^ 4) / (5 × 1)  or  (5 ^ 1) ^ (4 + (2 - 3))  or  (5 / (3 - 2)) ^ (4 - 1)
126 = (5 ^ (2 + 1)) + (4 - 3)  or  2 × ((4 ^ 3) - (1 ^ 5))  or  4 - ((2 - (5 ^ 3)) + 1)  or  4 + ((5 ^ (1 + 2)) - 3)  or  (3 × ((4 × 5) + 1)) × 2
127 = (4 / 2) + (1 × (5 ^ 3))  or  (4 - (2 - (5 ^ 3))) ^ 1  or  (4 + ((5 ^ 3) - 2)) ^ 1  or  ((5 × 1) ^ 3) + (4 / 2)  or  (5 ^ 3) + ((4 / 2) ^ 1)
128 = ((4 ^ 3) × 2) / (1 ^ 5)  or  (2 ^ (4 + 3)) × (1 ^ 5)  or  ((3 + 5) × (2 ^ 4)) ^ 1  or  1 + ((5 ^ 3) - (2 - 4))  or  (3 - 1) ^ (5 + (4 - 2))
129 = (1 ^ 5) + (2 × (4 ^ 3))  or  ((2 - 1) × (5 ^ 3)) + 4  or  ((1 + (2 ^ 5)) × 4) - 3  or  1 + (2 ^ ((4 × 3) - 5))  or  (2 ^ ((4 × 3) - 5)) + 1
130 = (3 + (5 ^ (4 - 1))) + 2  or  (5 × (1 + (3 × 4))) × 2  or  (5 ^ 3) + ((4 + 2) - 1)  or  5 × (2 × ((3 × 4) + 1))  or  (((2 ^ 5) × 4) - 1) + 3
131 = (4 × (2 ^ 5)) + (1 × 3)  or  ((4 + (5 ^ 3)) + 2) ^ 1  or  (1 × 3) + ((2 ^ 5) × 4)  or  5 - ((1 - (4 ^ 3)) × 2)  or  (1 × (5 ^ 3)) + (2 + 4)
132 = (2 ^ (4 + 3)) + (5 - 1)  or  5 + (2 + ((1 + 4) ^ 3))  or  2 + (1 + (4 + (5 ^ 3)))  or  ((5 ^ 3) + 4) + (1 + 2)  or  (((5 × 2) + 1) × 3) × 4
133 = ((5 ^ 3) + (4 × 2)) ^ 1  or  (5 ^ 1) + ((4 ^ 3) × 2)  or  (((4 ^ 3) × 2) + 5) / 1  or  ((2 ^ (4 + 3)) + 5) / 1  or  (5 + ((4 ^ 3) × 2)) × 1
134 = ((2 ^ (4 + 3)) + 1) + 5  or  (5 ^ 3) + ((4 × 2) + 1)  or  (2 ^ (3 + 4)) + (1 + 5)  or  ((4 ^ 3) × 2) + (1 + 5)  or  (5 + ((4 ^ 3) × 2)) + 1
135 = ((2 + 1) ^ 4) × (5 / 3)  or  5 × (((2 + 1) ^ 4) / 3)  or  5 / (3 / ((2 + 1) ^ 4))  or  3 × (5 × (1 + (2 × 4)))  or  ((2 × 4) + 1) × (5 × 3)
136 = (1 + (4 ^ 2)) × (5 + 3)  or  2 × (((5 - 1) ^ 3) + 4)  or  4 × (3 + ((2 ^ 5) - 1))  or  (1 + (4 ^ 2)) × (3 + 5)  or  2 × ((5 + (4 ^ 3)) - 1)
137 = (5 × (3 ^ (4 - 1))) + 2  or  ((3 ^ (4 - 1)) × 5) + 2  or  (4 × (2 + 1)) + (5 ^ 3)  or  2 - (5 × ((1 - 4) ^ 3))  or  (3 × 4) + (5 ^ (2 + 1))
138 = (2 × (5 + (4 ^ 3))) × 1  or  (((4 ^ 3) + 5) × 2) × 1  or  (1 × 2) × ((4 ^ 3) + 5)  or  (2 × ((4 ^ 3) + 5)) × 1  or  2 × (5 + ((4 / 1) ^ 3))
139 = ((5 + (4 ^ 3)) × 2) + 1  or  ((4 × 3) ^ 2) - (5 / 1)  or  1 × (((3 × 4) ^ 2) - 5)  or  (((2 ^ 5) + 3) × 4) - 1  or  4 + ((3 ^ (1 + 2)) × 5)
140 = (5 ^ 3) - (1 - (2 ^ 4))  or  ((5 ^ 3) - 1) + (4 ^ 2)  or  (5 × 4) × (1 + (3 × 2))  or  (((2 ^ 5) + 3) × 4) ^ 1  or  5 × (((3 × 2) + 1) × 4)
141 = ((3 + (2 ^ 5)) × 4) + 1  or  ((2 ^ 4) + (5 ^ 3)) / 1  or  (5 ^ 3) + ((1 × 2) ^ 4)  or  (2 ^ 4) + (5 ^ (3 / 1))  or  (5 ^ 3) + (4 ^ (2 ^ 1))
142 = ((4 ^ 2) + 1) + (5 ^ 3)  or  (4 ^ 2) + (1 + (5 ^ 3))  or  (1 + (4 ^ 2)) + (5 ^ 3)  or  (2 ^ 4) + (1 + (5 ^ 3))  or  ((5 ^ 3) + (2 ^ 4)) + 1
143 = (((4 + 3) + 5) ^ 2) - 1  or  ((4 + (5 + 3)) ^ 2) - 1  or  ((4 + (3 + 5)) ^ 2) - 1  or  ((5 + (4 + 3)) ^ 2) - 1  or  (((3 + 5) + 4) ^ 2) - 1
144 = (((3 + 5) + 4) ^ 2) × 1  or  (3 × ((5 + 1) × 4)) × 2  or  (5 + (4 + 3)) ^ (2 ^ 1)  or  (2 ^ (3 + 1)) × (4 + 5)  or  (1 + 5) × ((2 × 4) × 3)
145 = 1 + ((5 + (4 + 3)) ^ 2)  or  (1 ^ 5) + ((3 × 4) ^ 2)  or  (4 + 1) × ((2 ^ 5) - 3)  or  (1 ^ 5) + ((4 × 3) ^ 2)  or  1 + ((3 + (5 + 4)) ^ 2)
146 = ((3 ^ (2 + 4)) + 1) / 5  or  (1 + (3 ^ (2 + 4))) / 5  or  (1 + (3 ^ (4 + 2))) / 5  or  ((3 ^ (4 + 2)) + 1) / 5
147 = (((4 × 5) + 1) ^ 2) / 3  or  3 + (4 × ((5 + 1) ^ 2))  or  3 + (((1 + 5) ^ 2) × 4)  or  (((1 + 5) ^ 2) × 4) + 3  or  (((5 + 1) ^ 2) × 4) + 3
148 = ((3 × (1 - 5)) ^ 2) + 4  or  5 + (((3 × 4) ^ 2) - 1)  or  (((1 - 5) × 3) ^ 2) + 4  or  5 + (((4 × 3) ^ 2) - 1)  or  (((3 × 4) ^ 2) - 1) + 5
149 = (((3 × 4) × 1) ^ 2) + 5  or  (5 + ((3 × 4) ^ 2)) ^ 1  or  (5 + ((4 × 3) ^ 2)) ^ 1  or  ((4 × 3) ^ 2) + (5 × 1)  or  1 × (((3 × 4) ^ 2) + 5)
150 = 5 × ((4 + 1) × (3 × 2))  or  ((1 + 4) × 2) × (5 × 3)  or  ((4 × 3) ^ 2) + (5 + 1)  or  (2 × 5) × (3 × (1 + 4))  or  3 × ((5 × (1 + 4)) × 2)
151 = (2 × ((3 ^ 4) - 5)) - 1  or  (((3 ^ 4) - 5) × 2) - 1
152 = ((4 × 5) - 1) × (2 ^ 3)  or  2 × ((3 ^ (4 × 1)) - 5)  or  ((3 ^ 4) - 5) × (1 × 2)  or  (((3 ^ 4) - 5) × 2) × 1  or  (2 ^ 3) × ((5 × 4) - 1)
153 = 1 - (2 × (5 - (3 ^ 4)))  or  1 - ((5 - (3 ^ 4)) × 2)  or  (((3 ^ 4) - 5) × 2) + 1  or  1 + (((3 ^ 4) - 5) × 2)  or  (2 × ((3 ^ 4) - 5)) + 1
154 = ((1 - 5) + (3 ^ 4)) × 2  or  2 × (1 + ((3 ^ 4) - 5))  or  (((3 ^ 4) - 5) + 1) × 2  or  2 × ((1 + (3 ^ 4)) - 5)  or  ((3 ^ 4) + (1 - 5)) × 2
155 = (4 + ((1 + 2) ^ 3)) × 5  or  (2 × ((3 ^ 4) - 1)) - 5  or  ((3 ^ (1 + 2)) + 4) × 5  or  5 × (4 + ((1 + 2) ^ 3))  or  (4 + (3 ^ (1 + 2))) × 5
156 = ((3 ^ 4) × 2) - (1 + 5)  or  4 × (((5 + 1) ^ 2) + 3)  or  4 × (((1 + 5) ^ 2) + 3)  or  (((3 ^ 4) × 2) - 1) - 5  or  4 × ((5 × (2 ^ 3)) - 1)
157 = ((2 × (3 ^ 4)) - 5) / 1  or  ((3 ^ (4 / 1)) × 2) - 5  or  ((2 ^ (1 + 4)) × 5) - 3  or  ((3 ^ 4) / (1 / 2)) - 5  or  (((3 ^ 4) × 2) - 5) / 1
158 = (1 - 5) + ((3 ^ 4) × 2)  or  (((3 ^ 4) × 2) + 1) - 5  or  (2 × (3 ^ (5 - 1))) - 4  or  (1 + ((3 ^ 4) × 2)) - 5  or  ((2 × (3 ^ 4)) - 5) + 1
159 = (2 × (1 + (3 ^ 4))) - 5  or  (2 × ((3 ^ 4) + 1)) - 5  or  (5 / (2 / (4 ^ 3))) - 1  or  (((3 ^ 4) + 1) × 2) - 5  or  ((5 × (4 ^ 3)) / 2) - 1
160 = 1 × ((2 ^ 3) × (5 × 4))  or  (5 × (3 + 1)) × (2 × 4)  or  ((5 × (4 ^ 3)) × 1) / 2  or  4 × ((1 + 3) × (5 × 2))  or  5 / (2 / (4 ^ (1 × 3)))
161 = 1 + ((5 × 4) × (2 ^ 3))  or  ((3 ^ 4) × 2) - (1 ^ 5)  or  (5 × ((4 ^ 3) / 2)) + 1  or  ((5 × (2 ^ 3)) × 4) + 1  or  (2 × (3 ^ 4)) - (1 ^ 5)
162 = (3 - 1) × ((4 + 5) ^ 2)  or  (3 ^ 5) / ((2 / 4) + 1)  or  ((1 + 2) ^ 4) × (5 - 3)  or  (2 × ((4 - 1) ^ 5)) / 3  or  (3 ^ 4) × (5 - (2 + 1))
163 = 3 + (5 × (2 ^ (4 + 1)))  or  3 + ((2 ^ 5) × (1 + 4))  or  ((2 ^ (1 + 4)) × 5) + 3  or  3 + ((2 ^ 5) × (4 + 1))  or  ((3 ^ 4) × 2) + (1 ^ 5)
164 = ((1 ^ 5) + (3 ^ 4)) × 2  or  4 × (1 + (5 × (2 ^ 3)))  or  ((3 ^ 4) + (1 ^ 5)) × 2  or  ((1 + (3 × 4)) ^ 2) - 5  or  (((2 ^ 3) × 5) + 1) × 4
165 = (((3 ^ 4) - 1) × 2) + 5  or  5 × (1 + ((4 ^ 3) / 2))  or  (1 + ((4 ^ 3) / 2)) × 5  or  (2 × ((3 ^ 4) - 1)) + 5  or  ((4 × (2 ^ 3)) + 1) × 5
166 = 5 - (1 - ((3 ^ 4) × 2))  or  (2 × (3 ^ (5 - 1))) + 4  or  4 + (2 × (3 ^ (5 - 1)))  or  (2 / (3 ^ (1 - 5))) + 4  or  (5 - 1) + (2 × (3 ^ 4))
167 = (5 + (2 × (3 ^ 4))) ^ 1  or  ((2 × 1) × (3 ^ 4)) + 5  or  5 + (1 × (2 × (3 ^ 4)))  or  5 + ((3 ^ 4) × (2 / 1))  or  ((3 ^ 4) × 2) + (1 × 5)
168 = ((4 × 5) + 1) × (2 ^ 3)  or  (4 + 3) × ((5 ^ 2) - 1)  or  (1 + ((3 ^ 4) × 2)) + 5  or  ((5 ^ 2) - 1) × (3 + 4)  or  (2 × (3 ^ 4)) + (1 + 5)
169 = (1 + ((5 + 3) + 4)) ^ 2  or  ((3 + 4) + (1 + 5)) ^ 2  or  ((5 + (3 + 4)) + 1) ^ 2  or  ((1 + (3 ^ 4)) × 2) + 5  or  ((3 × 4) + (1 ^ 5)) ^ 2
170 = 2 × ((3 ^ 4) + (5 - 1))  or  (((3 ^ 4) - 1) + 5) × 2  or  2 × ((3 ^ 4) - (1 - 5))  or  ((3 ^ 4) - (1 - 5)) × 2  or  2 × (((3 ^ 4) - 1) + 5)
171 = (3 ^ 2) × ((4 × 5) - 1)  or  ((4 × 5) - 1) × (3 ^ 2)  or  ((2 ^ (5 + 4)) + 1) / 3  or  ((5 × 4) - 1) × (3 ^ 2)  or  (2 × ((3 ^ 4) + 5)) - 1
172 = 1 × (((3 ^ 4) + 5) × 2)  or  (1 × 2) × ((3 ^ 4) + 5)  or  2 × (((3 ^ 4) + 5) × 1)  or  (((3 ^ 4) + 5) × 2) / 1  or  1 × (2 × (5 + (3 ^ 4)))
173 = 1 + (2 × ((3 ^ 4) + 5))  or  1 + (((3 ^ 4) + 5) × 2)  or  1 + (2 × (5 + (3 ^ 4)))  or  (((3 ^ 4) + 5) × 2) + 1  or  1 + ((5 + (3 ^ 4)) × 2)
174 = (5 + (1 + (3 ^ 4))) × 2  or  ((4 + 3) × (5 ^ 2)) - 1  or  (((4 × 3) + 1) ^ 2) + 5  or  5 + ((1 + (4 × 3)) ^ 2)  or  (((3 ^ 4) + 1) + 5) × 2
175 = ((3 + 4) × (5 ^ 2)) ^ 1  or  1 × ((4 + 3) × (5 ^ 2))  or  ((3 + 4) × (5 ^ 2)) × 1  or  5 × (((3 ^ 2) × 4) - 1)  or  (1 + 4) × (3 + (2 ^ 5))
176 = 4 × (((3 ^ 2) × 5) - 1)  or  1 + ((4 + 3) × (5 ^ 2))  or  1 + ((5 ^ 2) × (4 + 3))  or  ((5 ^ 2) × (3 + 4)) + 1  or  ((3 + 4) × (5 ^ 2)) + 1

• Good job. Add the solution for zero [0-100] to have a complete answer. Commented Mar 22, 2017 at 0:07
• Nice job! I wonder what the maximum is!
– user35295
Commented Mar 22, 2017 at 1:18
• Awesome. But I recommend you (post-process) to a) sort the solutions from L-to-R to reward expressions using only simple arithmetic, and penalize ^ and ! Because 0 = (1+2-3) x (4+5) is much nicer than (3 - (1 + 2)) ^ (4 / 5). And (3 ^ 2) - (4 + (5 × 1)) is pretty good though. ...
– smci
Commented Mar 22, 2017 at 8:56
• ...and b) to remove the unnecessary nested parentheses due to commutativity in ((5 + 2) + 3). And (5 + (3 ^ 4)) can be simplified and made visually clearer as (5 + 3^4). Some of your exponentiation solutions are really hard to read...
– smci
Commented Mar 22, 2017 at 9:00
• @AllanCao the maximum will be something like 2^(3^(4^(5+1))) which is huge. The program has to be restricted from producing even mildly large numbers because it wouldn't be able to handle trying to calculate the exponentiations of them. Commented Mar 22, 2017 at 9:05