Treebeard's Stumper Answer

Crunching 1999
Here's a repeat of last year's winter vacation challenge. Use just the four digits 1, 9, 9, 9 in that order, with parentheses and any standard math operations, to make all the numbers from 1 to 100. For starters,
1 = 1+(99)x9 (+,, x, and /) 2 = (1^9)+(9/9) (^ is the power of) 3 = 1+sqrt(9)(9/9) (square root) 4 = 1+sqrt(9)!(9/9) (factorial, 3!=1x2x3=6) We've managed to get all the numbers using the digits of 1997 and 1998. Can we do it again this year?
The vacation challenge was to make the numbers 1 to 100 by adding math operations to the digits 1, 9, 9, 9 in that order. We did well, but 65 was difficult, and we were finally stumped by 68 despite all our tricks. These particular numbers seem so ordinary, but look closer. 65 is 5 times 13, and 68 is 4 times 17. These prime factors are hard to make with just 9s (and 3s and 6s) to work with. It's dangerous to claim that something can't be done, but that's my guess. The complete list of all the numbers we managed to get is below.
Notes:
We dedicated a whiteboard to this stumper at school. It helps to have many brains at work!
I gave up on 65 and 68. Then Graybear suggested allowing (.9...) = 1, which gives:
_ 65 = ((1+sqrt(9))^sqrt(9))+.9There has been endless debate about whether 0.999... equals 1 in the sci.math newsgroup. Offhand, it works just fine, though it's ugly in ASCII. I proposed this solution to the kids at school by arguing:If 1/3 = .333... and 2/3 = .666... then 3/3 = .999... = 1Or with a bit of algebra,10x = 9.999...  x = .999...  so 9x = 9 and x = 1The kids were reticent. I never spelled out exact rules for this stumper, but I've seen this challenge put forth like so:The digits all have to be used EXACTLY ONCE and IN THE ORDER GIVEN. They can be combined using any number of the operators +,  (negate and/or subtract), * (times), / (divide), ^ (power), factorial, square root, nth roots (but this uses the digit n), concatenate (but original digits only  so sqrt(9)8 does NOT make 38), decimal point (again on original digits only), repeater dots ... (e.g. .98... makes 98/99). Floor and ceiling functions are NOT allowed. Matching brackets are allowed.Accepting these rules, we have 65. The only number we can't get is 68.This is a silly stumper, but it's compelling! It's interesting to think of this problem as defining a function f( ). We feed it a year (actually any number), and it returns the first number that can't be made with those digits. So f(1997) and f(1998) are something greater than 100, and f(1999) equals 68. I suspect a graph of this function would show interesting regularities, but there's no easy way to get the data. This would be a good computer programming puzzle!
I'm sure this challenge is not possible for the year 2000, so f(2000) is much less than 100. (But what is it?) There's just not enough digits to work with, though it helps that 0!=1. Check your calculator, and don't ask me why! I'm curious when is the next year after 2000 when it can be done?
Here is the complete 1999 list of numbers from 0 to 100, as found by DMS students. Remember order of operations: PEMDAS  parentheses, exponents, multiply and divide, then add and subtract. It's useful that the square root of 9 is 3, and 3! is 1*2*3 = 6. 3! is "3 factorial", all the whole numbers up to 3 multiplied together. There's usually a "!" key on calculators. (We've been calling it "3 bang" at school.) We still haven't been able to get 68. It's frustrating to be so close, but it's still an impressive list!
0=1*(99)*9
1=1+(99)*9
2=1*(9+9)/9
3=1*sqrt(9)+99
4=1+99+sqrt(9)
5=1+sqrt(9)+(9/9)
6=1*(99)+sqrt(9)!
7=1+9+sqrt(9)sqrt(9)!
8=(1*9)(9/9)
9=1*9+(99)
10=1+9+99
11=1+9+(9/9)
12=1*9/sqrt(9)+9
13=1+9/sqrt(9)+9
14=1+9+9sqrt(9)
15=1*9+9sqrt(9)
16=1+9+9sqrt(9)
17=1+sqrt(9)+sqrt(9)!+9
18=(1+(9/9))*9
19=1+sqrt(9)!+sqrt(9)!+sqrt(9)!
20=1+(sqrt(9)*9)sqrt(9)!
21=1*sqrt(9)+9+9
22=1+9+9+sqrt(9)
23=1+(9*sqrt(9))sqrt(9)
24=1*(9*sqrt(9))sqrt(9)
25=1+sqrt(9)!+9+9
26=1+9+9+9
27=1*(9+9+9)
28=1+9+9+9
29=1+(sqrt(9)!*sqrt(9)!)sqrt(9)!
30=(1+9)*(9sqrt(9)!)
31=1+(sqrt(9)!*sqrt(9)!)sqrt(9)!
32=1+sqrt(9)!+(9*sqrt(9))
33=1*sqrt(9)!+(9*sqrt(9))
34=1+sqrt(9)!+(9*sqrt(9))
35=1+9+(9*sqrt(9))
36=(1*9)+(9*sqrt(9))
37=1+9+(9*sqrt(9))
38=1+(sqrt(9)!*sqrt(9)!)+sqrt(9)
39=(1+9)*sqrt(9)+9
40=1+(sqrt(9)!*sqrt(9)!)+sqrt(9)
41=1+(sqrt(9)!*sqrt(9)!)+sqrt(9)!
42=(1+sqrt(9))*9+sqrt(9)!
43=1+(sqrt(9)!*sqrt(9)!)+sqrt(9)!
44=1+(sqrt(9)!*sqrt(9)!)+9
45=1*(sqrt(9)!*sqrt(9)!)+9
46=1+(sqrt(9)!*sqrt(9)!)+9
47=1+(9*sqrt(9)!)sqrt(9)!
48=1*(9*sqrt(9)!)sqrt(9)!
49=1+(9*sqrt(9)!sqrt(9)!
50=1+(9*sqrt(9)!)sqrt(9)
51=1*(9*sqrt(9)!)sqrt(9)
52=1+(9*sqrt(9)!)sqrt(9)
53=1+(9+9)*sqrt(9)
54=1*(9+9)*sqrt(9)
55=1+(9+9)*sqrt(9)
56=1+(9*sqrt(9)!)+sqrt(9)
57=1*(9*sqrt(9)!)+sqrt(9)
58=1+(9*sqrt(9)!)+sqrt(9)
59=1+(9*sqrt(9)!)+sqrt(9)!
60=1*(9*sqrt(9)!)+sqrt(9)!
61=1+(9*sqrt(9)!)+sqrt(9)!
62=1+(9*sqrt(9)!)+9
63=1*(9*sqrt(9)!)+9
64=1+(9*sqrt(9)!)+9
65=((1+sqrt(9))^sqrt(9))+(.9...)
66=(1+9)*sqrt(9)!+sqrt(9)!
67=(1+sqrt(9))^sqrt(9)+sqrt(9)
68=
69=(1+9)*sqrt(9)!+9
70=(1+sqrt(9))^sqrt(9)+sqrt(9)!
71=1+(9*9)9
72=1*(9*9)9
73=1+(9*9)9
74=1+(9*9)sqrt(9)!
75=1*(9*9)sqrt(9)!
76=1+(9*9)sqrt(9)!
77=1+(9*9)sqrt(9)
78=1*(9*9)sqrt(9)
79=1+(9*9)sqrt(9)
80=1+(sqrt(9)*sqrt(9)*9)
81=1*(sqrt(9)*sqrt(9)*9)
82=1+(sqrt(9)*sqrt(9)*9)
83=1+sqrt(9)+(9*9)
84=1*sqrt(9)+(9*9)
85=1+sqrt(9)+(9*9)
86=1+sqrt(9)!+(9*9)
87=1*sqrt(9)!+(9*9)
88=1+sqrt(9)!+(9*9)
89=1+9+(9*9)
90=(1*9)+(9*9)
91=1+9+(9*9)
92=1sqrt(9)!+99
93=(1+9)*9+sqrt(9)
94=1+99sqrt(9)!
95=1sqrt(9)+99
96=(1+9)*9+sqrt(9)!
97=1sqrt(9)+99
98=(1^9)+99
99=((1+9)*9)+9
100=(1+9)*(9/.9)
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