# Create numbers from 1-100 using 1846

Create all numbers $$1$$ to $$100$$ using equations made up of $$1,4,6,8$$.

### Rules:

1. Use all four digits exactly once
2. Allowed operations: $$+,\,-,\,\times,\,\div,\,!$$ (factorial), exponentiation ($$a^b$$), square root ($$\sqrt{\text{ }}$$).
3. Parentheses and grouping (concatenation) (e.g. "$$21$$" from $$2$$ and $$1$$) are also allowed.
4. Exponentiation can only be used in the number order with the numbers provided. (eg. $$1^9+6+8$$ is allowed, but $$1^6+9+8$$ isn't).
5. The modulus operator ($$a\pmod b$$) is not allowed.
6. Rounding is not allowed (e.g. $$201/8=25$$).
7. Decimal points are allowed.
• Why is 9 in your examples when the digits are 1, 8, 4, and 6? Also, is the order constrained? This doesn't feel like a riddle, and the rules are kind of vague… Commented Nov 7, 2023 at 0:24
• What about square roots with the order of digits? Can I do $\sqrt{1+8}$? The order in the title does not match the order in the body. What is the correct order? Please make your examples apply, so $21$ does not consist of allowable digits. Commented Nov 7, 2023 at 3:20

Assuming there are typos in the examples given in the question.
My answer uses no decimals and no exponentiation.
Only two of the solutions use division.
Only two of the solutions use multiple-digit numbers.

Only $$74$$ and $$77$$ need multiple digits.
Only $$83$$ and $$85$$ need division.

$$1 = 4 + 6 - 8 - 1$$
$$2 = \sqrt{4} \times (6 - 1) - 8$$
$$3 = 1 + 4 - (8 - 6)$$
$$4 = (4 - 1)! - (8 - 6)$$
$$5 = 8 - 6 + 4 - 1$$
$$6 = (8 - 6) \times (4 - 1)$$
$$7 = 8 - 6 + 1 + 4$$
$$8 = (4 - 1)! + 8 - 6$$
$$9 = 6 - 4 + 8 - 1$$
$$10 = (4 - 1) \times 6 - 8$$
$$11 = 6 - 4 + 1 + 8$$
$$12 = (8 - 6 + 1) \times 4$$
$$13 = \sqrt{(1 + 8)} + 4 + 6$$
$$14 = (6 - 4) \times (8 - 1)$$
$$15 = 4 \times 6 - (1 + 8)$$
$$16 = (6 - 1) \times 8 - 4!$$
$$17 = 4 - 1 + 6 + 8$$
$$18 = (4 - 1) \times 8 - 6$$
$$19 = 1 + 4 + 6 + 8$$
$$20 = (1 + 6) \times 4 - 8$$
$$21 = 4!! - 1 + 6 + 8$$
$$22 = (8 - 1) \times 4 - 6$$
$$23 = (8 - 4) \times 6 - 1$$
$$24 = (6 - 4 + 1) \times 8$$
$$25 = 4 \times 8 - (1 + 6)$$
$$26 = (4 - 1) \times 6 + 8$$
$$27 = 4 \times 8 - (6 - 1)$$
$$28 = (6 - 1) \times 4 + 8$$
$$29 = (1 + 4)!! + 6 + 8$$
$$30 = (8 - 4 + 1) \times 6$$
$$31 = 8 - 1 + 4 \times 6$$
$$32 = (6 - 1) \times 8 - 4!!$$
$$33 = 1 + 8 + 4 \times 6$$
$$34 = (8 - 1) \times 4 + 6$$
$$35 = 6!! - (4 + 8) - 1$$
$$36 = (1 + 6) \times 4 + 8$$
$$37 = 6 - 1 + 4 \times 8$$
$$38 = (1 + 4) \times 6 + 8$$
$$39 = 1 + 6 + 4 \times 8$$
$$40 = (1 + 6 - \sqrt{4}) \times 8$$
$$41 = 6 \times 8 - 4!! + 1$$
$$42 = (1 + 8) \times 4 + 6$$
$$43 = 6 \times 8 - (1 + 4)$$
$$44 = (6 - 1) \times 8 + 4$$
$$45 = 6 \times 8 - (4 - 1)$$
$$46 = (1 + 4) \times 8 + 6$$
$$47 = (6 - 1)!! + 4 \times 8$$
$$48 = (6 - 4 + 1)! \times 8$$
$$49 = 6 \times 8 - 1 + \sqrt{4}$$
$$50 = (1 + 8) \times 6 - 4$$
$$51 = 4 - 1 + 6 \times 8$$
$$52 = (6 - 1 + 8) \times 4$$
$$53 = 1 + 4 + 6 \times 8$$
$$54 = (4 - 1)! + 6 \times 8$$
$$55 = (6 + 8) \times 4 - 1$$
$$56 = (\sqrt{4} + 6 - 1) \times 8$$
$$57 = (6 + 8) \times 4 + 1$$
$$58 = (1 + 8) \times 6 + 4$$
$$59 = 4 - 1 + 8 + 6!!$$
$$60 = (1 + 6) \times 8 + 4$$
$$61 = 1 + 4 + 8 + 6!!$$
$$62 = (1 + 8) \times 6 + 4!!$$
$$63 = (1 + 4)!! + 6 \times 8$$
$$64 = (6 - 1) \times 8 + 4!$$
$$65 = (\sqrt{4} + 6) \times 8 + 1$$
$$66 = (4 - 1 + 8) \times 6$$
$$67 = \sqrt{(8!! - 4! + 1)} + 6!!$$
$$68 = (6 - 1)!! \times 4 + 8$$
$$69 = 4!! \times 8 + 6 - 1$$
$$70 = (1 + 4) \times (6 + 8)$$
$$71 = (4 + 8) \times 6 - 1$$
$$72 = (4 - 1 + 6) \times 8$$
$$73 = (4 + 8) \times 6 + 1$$
$$74 = (4 - 1)! + 68$$
$$75 = (8 - 1)!! - (4! + 6)$$
$$76 = (8 - 1) \times 4 + 6!!$$
$$77 = \sqrt{841} + 6!!$$
$$78 = (1 + 4 + 8) \times 6$$
$$79 = (4 + 6) \times 8 - 1$$
$$80 = \sqrt{4} \times (6 - 1) \times 8$$
$$81 = (4 + 6) \times 8 + 1$$
$$82 = (1 + 4)!! \times 6 - 8$$
$$83 = \frac{6!}{8} - (4!! - 1)$$
$$84 = (1 + 6) \times (4 + 8)$$
$$85 = \frac{6!}{8} - (1 + 4)$$
$$86 = (6!! - 1) \times \sqrt{4} - 8$$
$$87 = \sqrt{4} \times 6!! - (1 + 8)$$
$$88 = (1 + 4 + 6) \times 8$$
$$89 = \sqrt{4} \times 6!! - (8 - 1)$$
$$90 = (1 + 8) \times (4 + 6)$$
$$91 = (8 - 1)!! - (4!! + 6)$$
$$92 = ((6 - 1)!! + 8) \times 4$$
$$93 = (1 + 6)!! - (4 + 8)$$
$$94 = (6 \times 8 - 1) \times \sqrt{4}$$
$$95 = \sqrt{4} \times 6 \times 8 - 1$$
$$96 = ((4 - 1)! + 6) \times 8$$
$$97 = \sqrt{4} \times 6 \times 8 + 1$$
$$98 = (1 + 4)!! \times 6 + 8$$
$$99 = (1 + 6)!! - 8 + \sqrt{4}$$
$$100 = ((\sqrt{(1 + 8)})!)!! + 6!! + 4$$

• I think that this is a very good answer! Also good job on achieving the result without grouping/concatenation! Commented Nov 10, 2023 at 16:52
• @CrSb0001 thank you, but two of them do use grouping. I side-stepped the sequence issue by not using powers. I also tried another exercise with 1, 4, 6, 8 always in that sequence, but only solved 75 of the 100 that way. Commented Nov 10, 2023 at 17:05