# Create all numbers from 1-100 by using 1,3,3,7

Create all the numbers from $$1$$ to $$100$$ using the numbers $$1,3,3,$$ and $$7.$$

You can only use each number once, except for the $$3,$$ of which you have two.
You can use addition $$(x+y),$$ subtraction $$(x−y),$$ division $$(\frac{x}{y}),$$ multiplication $$(x\times y),$$ exponentiation $$(x^y),$$ roots $$(y\sqrt x)$$, factorials $$(x!)$$ and ceiling and floor $$(\lceil x\rceil,\lfloor y\rfloor)$$.
You can combine numbers like $$1$$ and $$7$$ to $$17$$ etc.
Use of any types of brackets are also allowed.
Good luck!

• Hi @H_D I'll try this ;) +1 for your first post ;) – Omega Krypton Jan 29 at 11:28
• We are not allowed to use brackets? :( – Tweakimp Jan 29 at 11:46
• Use of brackets are allowed. Don't be sad. – H_D Jan 29 at 12:07
• Why the downvote? This site should not be discriminating new members of our community, please check out the Code of Conduct. We should all be community-minded and not simply drown new members with -1s. This is a legitimate question, where I do not see the point of downvoting. – Omega Krypton Jan 29 at 12:40
• I agree with both of @OmegaKrypton's comments, although adding such a restriction after there are already 3 answers is probably bad form. If we're talking restrictions, I would love to see concatenation banished from these puzzles in general. – tilper Jan 29 at 12:57

1 number missing. I give up

$$0 = 7-3-3-1$$
$$1 = 7 \times 1 - 3 - 3$$
$$2 = 7 + 1 - 3 - 3$$
$$3 = \frac{7 - 1 + 3}{3}$$
$$4 = \sqrt{17-\frac{3}{3}}$$
$$5 = 3 \times (3 + 1) - 7$$
$$6 = 7 - 1 + 3 - 3$$
$$7 = 7 \times 1 - 3 + 3$$
$$8 = 7 + 1 - 3 + 3$$
$$9 = 13 + 3 - 7$$
$$10 = 3^3 - 17$$
$$11 = 17 - 3 - 3$$
$$12 = 3 \times (7 - 3) \times 1$$
$$13 = 3 \times (7 - 3) + 1$$
$$14 = 7 \times (1 + \frac{3}{3})$$
$$15 = 3 \times (7 - 3 + 1)$$
$$16 = 17 - \frac{3}{3}$$
$$17 = 17 - 3 + 3$$
$$18 = 17 + \frac{3}{3}$$
$$19 = 7 \times 3 + 1 - 3$$
$$20 = 3^3 - 7 \times 1$$
$$21 = 3^3 - 7 + 1$$
$$22 = (7-3)! - 3 + 1$$
$$23 = 7 \times 3 + 3 - 1$$
$$24 = \frac{73 - 1}{3}$$
$$25 = 33 - 7 - 1$$
$$26 = 33 - 7 \times 1$$
$$27 = 33 - 7 + 1$$
$$28 = 3 \times 7 + 3! + 1$$
$$29 = 3! \times 3! - 7 \times 1$$
$$30 = 37 - 3! - 1$$
$$31 = 37 - 3! \times 1$$
$$32 = 37 - 3! + 1$$
$$33 = 37 - 3 - 1$$
$$34 = 37 - 3 \times 1$$
$$35 = 37 - 3 + 1$$
$$36 = (7-1) \times (3+3)$$
$$37 = 3! \times (3! - 1) + 7$$
$$38 = 3! \times 7 - 3 - 1$$
$$39 = 3! \times 7 - 3 \times 1$$
$$40 = 3! \times 7 - 3 + 1$$
$$41 = 37 + 3 + 1$$
$$42 = 37 + 3! - 1$$
$$43 = 37 + 3! \times 1$$
$$44 = 37 + 3! + 1$$
$$45 = 3! \times 7 + 3 \times 1$$
$$46 = 3! \times 7 + 3 + 1$$
$$47 = 3! \times 7 + 3! - 1$$
$$48 = 3! \times 7 + 3! \times 1$$
$$49 = 7 \times (3+3+1)$$
$$50 = (7+3) \times (3!-1)$$
$$51 = 17 \times (3! - 3)$$
$$52 = 13 \times (7-3)$$
$$53 = \lfloor\sqrt{7^3}\rfloor \times 3 - 1$$
$$54 = 17 + 3^3$$
$$55 = 7 \times 3 + (3+1)!$$
$$56 = (3\times 3 - 1) \times 7$$
$$57 = 17 \times 3 + 3!$$
$$58 = \lfloor\frac{7^3}{3!}\rfloor + 1$$
$$59 = 3! \times (7+3) - 1$$
$$60 = 3! \times (7+3) \times 1$$
$$61 = 3! \times (7+3) + 1$$
$$62 = 7\times 3^3 - 1$$
$$63 = 7\times 3^3 \times 1$$
$$64 = 7\times 3^3 + 1$$
$$65 = (7+3!) \times (3!-1)$$
$$66 = 7\times 3! + (3+1)!$$
$$67 = (3+1)^3 + \lceil\sqrt{7}\rceil = 64 + 3$$
$$68 = \lfloor\frac{7^3}{3!-1}\rfloor$$
$$69 = \lceil\frac{7^3}{3!-1}\rceil$$
$$70 = (7 + 3) \times (3! + 1)$$
$$71 = 71 -3 + 3$$
$$72 = (3! + 3!) \times (7-1)$$
$$73 = 73 + \lfloor\frac{1}{3}\rfloor$$
$$74 = 3^{3+1} - 7$$
$$75 = 73 + 3 - 1$$
$$76 = 73 + 3 \times 1$$
$$77 = 73 + 3 + 1$$
$$78 = 3! \times (7 + 3!) \times 1$$
$$79 = 3! \times (7 + 3!) + 1$$
$$80 = 3^{7-3} - 1$$
$$81 = 3^{7-3} \times 1$$
$$82 = 3^{7-3} + 1$$
$$83 = 7 \times (3! + 3!) - 1$$
$$84 = 7 \times (3! + 3!) \times 1$$
$$85 = 7 \times (3! + 3!) + 1$$
$$86 = 73 + 13$$
$$87 = \lceil{\sqrt{\sqrt{13^7}}}\rceil - 3 = \lceil{89.00222..}\rceil - 3$$
$$88 = 3^{3+1} + 7$$
$$89 = 13 \times 7 - \lceil\sqrt{3}\rceil$$
$$90 = 13 \times 7 - \lfloor\sqrt{3}\rfloor$$
$$91 = (3!+3!+1) \times 7$$
$$92 = 13 \times 7 + \lfloor\sqrt{3}\lfloor$$
$$93 = 13 \times 7 + \lceil\sqrt{3}\lceil$$
$$94 = 13 \times 7 + 3$$
$$95 =$$
$$96 = 17 \times 3! - 3!$$
$$97 = 73 + (3+1)!$$
$$98 = 71 + 3^3$$
$$99 = (3! - 1)! - 3\times 7$$
$$100 = 31 \times 3 + 7$$

Bonus:

If we allow log functions we can generate every number like this:
$$x = \log_{\frac{1}{\lfloor\sqrt7\rfloor}}\left({\log_3\underbrace{\sqrt{\sqrt{\dots\sqrt{3\,}\,}\,}}_\text{x square roots}}\right)$$
This is equivalent to
$$x = \log_{\frac12}\left({\log_3{3^{\frac{1}{2^x}}}}\right)$$
Going further:
$$x = \log_{\frac12}\left({\frac{1}{2^x}}\right)$$

• I think for 78 and 79, you meant $3!\times (7+3!)\times 1$ and $3!\times (7+3!)+1$, right? – H_D Jan 29 at 14:18
• You're welcome. It's kind of my responsibility to do so I guess? – H_D Jan 29 at 14:21
• Very close! You can do it!! – H_D Jan 29 at 15:42
• Here's a silly one for you: $95: 1+ \lfloor \sqrt{ \sqrt{ \sqrt{ \lfloor \sqrt{ 337 } \rfloor ! } } } \rfloor$ – Bass Jan 29 at 18:09

Trying my best to arrange the numbers in order:

1-10:

1= $$1*-(3+3-7)$$
2= $$1-(3+3-7)$$
3= $$13-3-7$$
4= $$1*floor(\sqrt{3})*-(3-7)$$
5= $$1^3*3+floor(\sqrt{7})$$
6= $$13*floor(\sqrt{3})-7$$
7= $$13+floor(\sqrt{3})-7$$
8= $$13-3-floor(7)$$
9= $$1*3^{\sqrt{(-3+7)}}$$
10= $$1+3^{\sqrt{(-3+7)}}$$

11-20:

11= $$13-floor(\sqrt{3})*floor(\sqrt{7})$$
12= $$13+floor(\sqrt{3})-floor(\sqrt{7})$$
13= $$13+ceil(\sqrt{3})-floor(\sqrt{7})$$
14= $$(1+3)!-3-7$$
15= $$13+floor(\sqrt{3})*floor(\sqrt{7})$$
16= $$(1+3)^{\sqrt{(-3+7)}}$$
17= $$13-3+7$$
18= $$1*3*3*floor(\sqrt{7})$$
19= $$13+3*floor(\sqrt{7})$$
20= $$(-1+3)*(3+7)$$

21-30:

21= $$1-floor(\sqrt{3})+3*7$$
22= $$1*floor(\sqrt{3})+3*7$$
23= $$1+floor(\sqrt{3})+3*7$$
24= $$1*3+3*7$$
25= $$(-1+3+3)^{floor(\sqrt{7})}$$
26= $$13*floor(\sqrt{3})*floor(\sqrt{7})$$
27= $$1*3*3*ceil(\sqrt{7})$$
28= $$-1+3^3+floor(\sqrt{7})$$
29= $$-1+3*(3+7)$$
30= $$1*3*(3+7)$$

31-40:

31= $$1+3*(3+7)$$
32= $$1*3^{3+floor(\sqrt{7})}$$
33= $$1+3^{3+floor(\sqrt{7})}$$
34= $$1*(-3)+37$$
35= $$1-3+37$$
36= $$-1*floor(\sqrt{3})+37$$
37= $$1*floor(\sqrt{3})+37$$
38= $$1+floor(\sqrt{3})+37$$
39= $$-1+3+37$$
40= $$1*3+37$$

41-50:

41= $$1+3+37$$
42= $$1*(3+3)*7$$
43= $$1+(3+3)*7$$
44= $$1+3!+37$$
45= $$(1+3)!+3*7$$
46= $$13*3+7$$
47= $$-1+3!+3!*7$$
48= $$(1+3)!*floor(\sqrt{3})*floor(\sqrt{7})$$
49= $$(1+3+3)*7$$
50= $$(-1+3!)*(3+7)$$

51-60:

51= $$1*floor(\sqrt{3!!})+floor(\sqrt{\sqrt{3!!}})^{floor(\sqrt{7})}$$
52= $$1+floor(\sqrt{3!!})+floor(\sqrt{\sqrt{3!!}})^{floor(\sqrt{7})}$$
53= $$1+ceil(\sqrt{3!!})+floor(\sqrt{\sqrt{3!!}})^{floor(\sqrt{7})}$$
54= $$1*3*3*floor(\sqrt{7})!$$
55= $$(-1+3!)(floor(\sqrt{\sqrt{3!!}}+ceil(\sqrt{7})!)$$
56= $$(-1+3!+3)*7$$
57= $$(-1+floor(\sqrt{3!!}))*ceil(\sqrt{3})+7$$
58= $$-(1+3)*3+floor(\sqrt{7!})$$
59= $$-13+ceil(\sqrt{3})+floor(\sqrt{7})$$
60= $$-13+3+floor(\sqrt{7!})$$

61-70:

61= $$-1*3*3+floor(\sqrt{7!})$$
62= $$-(1+3)*ceil(\sqrt{3})+floor(\sqrt{7!})$$
63= $$1*3*3*7$$
64= $$1+3*3*7$$
65= $$-1*3-ceil(\sqrt{3})+floor(\sqrt{7!})$$
66= $$-1-3-ceil(\sqrt{3})+floor(\sqrt{7!})$$
67= $$-1*3-floor(\sqrt{3})+floor(\sqrt{7!})$$
68= $$-1-floor(\sqrt{3})*floor(\sqrt{3})+floor(\sqrt{7!})$$
69= $$-1*3/3+floor(\sqrt{7!})$$
70= $$1*3/3*floor(\sqrt{7!})$$

71-80:

71= $$1*3/3+floor(\sqrt{7!})$$
72= $$-1+3*floor(\sqrt{3})+floor(\sqrt{7!})$$
73= $$1*3*floor(\sqrt{3})+floor(\sqrt{7!})$$
74= $$1+3*floor(\sqrt{3})+floor(\sqrt{7!})$$
75= $$-1+3*ceil(\sqrt{3})+floor(\sqrt{7!})$$
76= $$1*3*ceil(\sqrt{3})+floor(\sqrt{7!})$$
77= $$1+3*ceil(\sqrt{3})+floor(\sqrt{7!})$$
78= $$-1+3*3+floor(\sqrt{7!})$$
79= $$1*3*3+floor(\sqrt{7!})$$
80= $$1+3*3+floor(\sqrt{7!})$$

81-90:

81= $$-1+3!+3!+floor(\sqrt{7!})$$
82= $$1*3!+3!+floor(\sqrt{7!})$$
83= $$1+3!+3!+floor(\sqrt{7!})$$
84= $$-1+3^ceil(\sqrt{3})+floor(\sqrt{7!})$$
85= $$1*3^ceil(\sqrt{3})+floor(\sqrt{7!})$$
86= $$1+3^ceil(\sqrt{3})+floor(\sqrt{7!})$$

Extending Omega Krypton's answer with 8 to 15, keeping 1,3,3,7 in order. I use $$\circ$$ to display concatenation:

$$8 = 1+3-3+7$$
$$9 = \lceil 1* \sqrt[3]{3}\rceil+7$$
$$10 = 1 \circ \lfloor 3/3 \circ 7 \rfloor$$
$$11 = 1\circ 3- \lceil \sqrt[3]{7} \rceil$$
$$12 = 1\circ 3- \lceil 3/7 \rceil$$
$$13 = 1\circ 3+ \lfloor 3/7 \rfloor$$
$$14 = 1\circ 3+ \lceil 3/7 \rceil$$
$$15 = 1\circ 3+ \lceil \sqrt[3]{7} \rceil$$

This puzzle is a bit too broad, so I only have three example numbers below. I'm sure you can figure out what the rest might look like.

$$36 = \lfloor{\sqrt{1337}}\rfloor$$

$$11 = \lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{(\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{(\lfloor{\sqrt{1337}}\rfloor)!}}\rfloor}}\rfloor}}\rfloor}}\rfloor}}\rfloor)!}}\rfloor}}\rfloor}}\rfloor}}\rfloor$$
$$79 = \lfloor{\sqrt{\lfloor{\sqrt{(\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{(\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{\lfloor{\sqrt{(\lfloor{\sqrt{1337}}\rfloor)!}}\rfloor}}\rfloor}}\rfloor}}\rfloor}}\rfloor)!}}\rfloor}}\rfloor}}\rfloor}}\rfloor)!}}\rfloor}}\rfloor$$

• Surely I didn't think of this possibility. Well, fascinating. – H_D Jan 29 at 14:22

Work in progress, I'm trying to avoid using ceil, floor and root, so far so good but I don't think it's possible for some big numbers. I could have made some mistakes though.

1-10:

$$1=(1+3+3)/7$$
$$2=1+7-3-3$$
$$3=3*3-7+1$$
$$4=(31-3)/7$$
$$5=3!+3!-7/1$$
$$6=37-31$$
$$7=7+(3-3)*1$$
$$8=17-3*3$$
$$9=7+1+3/3$$
$$10=3!*3-7-1$$

11-20:

$$11=17-3-3$$
$$12=3!*3-7+1$$
$$13=7+3+3/1$$
$$14=7+3+3+1$$
$$15=7*3-3!/1$$
$$16=33-17$$
$$17=17-3+3$$
$$18=3*7-3/1$$
$$19=3*7-3+1$$
$$20=3!+3!+7+1$$

21-30:

$$21=31-7-3$$
$$22=17+3!-3$$
$$23=17+3+3$$
$$24=37-13$$
$$25=3*7+3+1$$
$$26=3!*3+7+1$$
$$27=(7-3)!+3/1$$
$$28=(7-3)!+3+1$$
$$29=3!*3!-7/1$$
$$30=3!*3!-7+1$$

31-40:

$$31=7*(3+1)+3$$
$$32=13*3-7$$
$$33=37-3-1$$
$$34=37-3/1$$
$$35=37-3+1$$
$$36=(7-1)*(3+3)$$
$$37=13+(7-3)!$$
$$38=71-33$$
$$39=37+3-1$$
$$40=37+3/1$$

41-50:

$$41=37+3+1$$
$$42=73-31$$
$$43=7*(3+3)+1$$
$$44=7*3!+3-1$$
$$45=7*3!+3/1$$
$$46=13*3+7$$
$$47=7*3!+3!-1$$
$$48=(7+1)*(3+3)$$
$$49=7*(3+3+1)$$
$$50=(7+3)*3!-1$$

51-60: (WIP)

$$51=$$
$$52=$$
$$53=$$
$$54=17*3+3$$
$$55=$$
$$56=$$
$$57=$$
$$58=$$
$$59=$$
$$60=(13+7)*3$$

61-70: (WIP)

$$61=$$
$$62=7*3*3-1$$
$$63=7*3*3/1$$
$$64=7*3*3+1$$
$$65=$$
$$66=$$
$$67=$$
$$68=37+31$$
$$69=73-3-1$$
$$70=(73-3)/1$$

71-80: (WIP)

$$71=73-3+1$$
$$72=71+3/3$$
$$73=$$
$$74=$$
$$75=$$
$$76=$$
$$77=71+3+3$$
$$78=$$
$$79=$$
$$80=$$

81-90: (WIP)

$$81=$$
$$82=$$
$$83=$$
$$84=$$
$$85=7*13-3!$$
$$86=31*3-7$$
$$87=$$
$$88=7*13-3$$
$$89=$$
$$90=$$

91-100 (WIP)

$$91=$$
$$92=$$
$$93=$$
$$94=7*13+3$$
$$95=$$
$$96=$$
$$97=7*13+3!$$
$$98=$$
$$99=$$
$$100=31*3+7$$