# 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)? – F1Krazy Mar 20 '17 at 14:55
• The original problem said that we can not use "glue" and combine digits or this would be really easy... – user35295 Mar 20 '17 at 14:58
• Should I delete this? So sorry, I am new. – user35295 Mar 20 '17 at 14:58
• don't because I already started on it :d. You can add that "no glue" restriction also. – Marius Mar 20 '17 at 14:59
• The question is awesome, and deserves to be like a new FizzBuzz test that includes recursion! – smci Mar 22 '17 at 9:02

## 4 Answers

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 Mar 21 '17 at 1:24
• For the less math-ept - would you mind running me through this solution? – Shadow Mar 21 '17 at 4:37
• What operation is !!? I had assumed factorial - factorial, but this makes for a very large number. – James Webster Mar 21 '17 at 9:57
• @JamesWebster en.wikipedia.org/wiki/Double_factorial – Taemyr Mar 21 '17 at 10:59
• This is both brilliant and beautiful. – ABcDexter Mar 21 '17 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}

Someone please double check it.

• You are quicker. Haha :) – Maria Deleva Mar 20 '17 at 15:05
• Just leave me a few more minutes. – Marius Mar 20 '17 at 15:05
• I'm done. Can someone please double check it. – Marius Mar 20 '17 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. :) – Maria Deleva Mar 20 '17 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. – Marius Mar 20 '17 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. – Marius Mar 20 '17 at 16:04
• Oops copy pasting error - I did have it. – Maria Deleva Mar 20 '17 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. – Maria Deleva Mar 20 '17 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 = [
('+', add),
('-', 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)}
answer = defaultdict(set)
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
answer[result_num].add(result_string)

return answer

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), '=',
'  or  '.join(sample(answer[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. – Tracy Cramer Mar 22 '17 at 0:07
• Nice job! I wonder what the maximum is! – user35295 Mar 22 '17 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 Mar 22 '17 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 Mar 22 '17 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. – Alex Hall Mar 22 '17 at 9:05