Math: Antares Expressions

[Monte314]

Recently posted math problems have been solved almost instantly, so I've cooked one up that has three levels of difficulty.

There are two "easy questions", two "hard questions", and three "very hard" questions. The very hard questions are probably best handled by writing a computer program (that's how I did them).

Consider expressions of the form A#B#C#D#E, where:

1.) Each letter A, B, C, D, and E is a positive digit 1, 2, 3, 4, 5, 6, 7, 8, 9
(duplications are allowed, so, for example, A and B could be the same digit.)

and

2.) # is one of the operations +, -, or * (addition, subtraction, multiplication).

For definiteness, we establish the convention that the operations are performed left-to-right, i.e., A#B#C#D#E = (((A#B)#C)#D)#E (as with the digits, reuse of operations is allowed.)

For example, we can obtain the value -11 as:

2-4*2-9+2 = (((2-4)*2)-9)+2 = ((-2*2)-9)+2 = (-4-9)+2 = -13+2 = -11
(We can also obtain it, for example, as 6-7*5*2-1)

There are 9*3*9*3*9*3*9*3*9 = 4,782,969 distinct expressions that arise in this way.

Many values can be obtained in multiple ways. For example, there are 55,009 distinct expressions that give 1!

I am calling these Antares Expressions, after the INTJ Forum's home-grown genius.

Easy Questions:
1.) What is the smallest value that can be obtained as an Antares Expression? (Hint: it will be negative)

2.) What is the largest value that can be obtained as a Antares Expression? (Hint: it will be positive)

Hard Questions:
3.) Can 2008 be obtained as an Antares Expression?

4.) Can your birth year be obtained as an Antares Expression? (Monte's can... in 132 diferent ways!)

Very Hard Questions:
5.) What specific value can be obtained from the largest number of different Antares Expressions? (Hint: It isn't 1. In fact, there are 16 values that occur more frequently than 1!)

6.) What is the smallest positive value that can only be obtained from one Antares Expression?

7.) What is the smallest positive value that cannot be obtained as an Antares Expression at all?

[TheLastMohican]

Okay, I can handle the "easy" questions (love the hints!):


1. (1-9)*9*9*9 = -5,832
2. 9*9*9*9*9 = 59,049


...And the "hard" questions:


3. 9*8*7*4-8 = 2008
4. 9*9*5-7*5 = 1990


I'll let the "very hard" questions bother my Ni for a while until the answers magically appear. That might take a decade or two.

[Monte314]

OK, TLM gets credit for the two easy problems! Yay!

On to 3 - 7...

Don't forget that you can also use addition and subtraction, not just multiplication!

[TheLastMohican]

Don't forget that you can also use addition and subtraction, not just multiplication!

*Smacks forehead* Right, I need to revise those.

EDIT: I edited my first post with the solutions.

[Wufnu]

I don't quite have the time to make programs for the very hard problems but I can lay out some thoughts about them for whoever may solve them.

For the first one, we want the most expressions. This means we are looking for a number that has as many single digit factors as possible but also be small enough so that addition and subtraction may be used. Just as an example, lets take 22. 2 * 10 + 2-1+1, 2 * 11 + 2 - 2 *1, etc etc. However, in this case, 11 has no factors which is a bad thing. In this case, I'm going to make a guess at 32 (2^5).

Problem 6:
For the second one, we are looking for the smallest possible value only obtainable by one Antares Expression. This means any single digit values are right out the door. This also suggests the multiple of any two prime values must be greater or equal to 11 as any non-prime number has a multiple made of prime numbers. This also suggests that each value, following from left to right, must be a prime number and must not be replaceable by another multiple of two numbers. For example, if the first two values were 5+3, then you could easily replace that with 4+4, etc. Since we're looking for the lowest value, we want to consider addition. For the most part, any number below 17 can be reproduced via addition or multiplication in more than one way and the sum must be double digits else it could be replaced by value * 1. Also, lets expand a little: if we had 17 as the value of the first two digits, any subtraction from his means that the first set of two values could be replaced by a lower number suggesting multiple sets of equal value. It should be apparent that, also, any addition to 17 would be replaceable by another set. We are looking for a multiple and it must be low number. 2 will not work, nor will 1 (assuming 1*2*3=2*1*3, ie: same expression). 3 will not work (51 = 9+8*3 = 9*5+6). In fact, we're looking for a number greater than 90. 102 fits the bill. We're now at (9+8)*6 = 102 which is, I believe, the smallest positive integer which can be created from 3 numbers.


This logic is wrong, leaving it to look at:
We're now looking for a number larger than 738 = 9*9*9+9. 816 = (9+8)*6*8 seems to fit this bill. Following the same logic, we need a number larger than 9*9*9*9+9 = 6570. Also, I have just realized that this logic is incorrect and now I'm disgusted with the whole process.

102 * 7 = 714 which obviously has multiples of 7 and 6. It also has a multiple of 3 however this results in a number of 238 whose largest single digit factor is 7, which results in 34 which is impossible to achieve with two single digit numbers. We are now looking for that last digit.


(9+8) *6 * 7 * 7 = 4998 seems to work for problem 6. Not sure if it's the lowest.

For problem 7, who knows. It will likely be a prime number plus or minus 10. From the first 3 digits alone, we know it will be greater than 90.
5146 = 2 * ([9*2*(9*8*2-1)]-1) certainly works but I doubt it's the lowest.






Wufnu added to this post, 64 minutes and 44 seconds later...


As I lay in bed, trying to sleep, I got onto something. We know that with 3 numbers we can reach all numbers up to 90. Also, (9+9)*5 = 90. The remaining two numbers can be added to equal any number between 0 and 18, essentially making a base 18 numbering system.
(9+9)*5+1-1 = 90
(9+9)*5+2-1 = 91
etc
(9+9)*5+9+9 = 108
(9+9)*6+1-1 = 108
(9+9)*6+2-1 = 109
etc etc until
(9+9)*9+9+9 = 180
9*4*5+1-1 = 180
9*4*5+2-1 = 181
9*4*5+1+1 = 182
etc until
9*4*5+1+8 = 189
9*4*5+2+8 = 190
9*4*5+2+9 = 191
etc until
9*4*5+9+9 =198
switching to base 9 number system: 22 * 9 + 1 = 199
((5*4)+2)*9+1 = 199
((5*4)+2)*9+9 = 207
((5*4)+3)*9+1 = 208 = 23*9 + 1
etc until
((5*4)+5)*9 +1 = 226 = 5*5*9+2-1
5*5*9+1+1 = 227
this continues until you have
(9*9+9) * 9 + 9= 819 = 9 * 91
820 = (9*4 + 5) * 5 *4

For 821 to exist within the rules, no number 821 +/- 9 must be found for any four numbers.

9*9*9 = 729 = 81*9
91*9 = 819 and 92*9 = 827 which would be available within 4 numbers if 91 were possible with 3 numbers, which it isn't. This would imply that 819 is impossible to reach with 4 numbers.

822 = 6*137 and for 822 to be available within 4 numbers, 137 must be available within 3 numbers. It isn't. The closest would be 136 = (9+8)*8 so 137 = (9+8)*8 + 1 which would make 822 = ((9+8) * 8 + 1) * 6

I'm going to change my answer for 7 to 821.

Again, probably wrong but once I started thinking in alternative base numbering systems I had to work it out.

[Monte314]

Yay! TLM has solved problems 3 and 4 by providing one of the *60* ways in which 2008 can be obtained as an Antares Expression, and doing his own birthyear (1990).

wufnu has proposed a brilliant method of analysis for generating solutions to 5 - 7. He is on to something here...

By the way: 821 is represented by 16 distinct Antares Expressions, among them (((9+8)*8)*6)+5.

[Tenacious B]

Wow, she already has her own expression. That is quite an accomplishment.;)

[TheLastMohican]

Wow, she already has her own expression. That is quite an accomplishment.;)

Indeed. I nominate Wufnu for the next one.

[Wufnu]

Hah, that won't work. Antares' name rolls off the tongue and has a nice sound to it. Mine sounds like a nerd sneezing. It'll never work.

[SiMey]

Zzzzzz huh what?

I used to like maths but gave it up years ago.

Don't make me think too much.

[ssrprotege]

Indeed. I nominate Wufnu for the next one.

I nominate TLM the next. TLM Expression doesn't sound too bad.

[TheLastMohican]

Mine sounds like a nerd sneezing.

:laugh: Now that "pronunciation" will be forever stuck in my head.

I nominate TLM the next. TLM Expression doesn't sound too bad.

Thanks. It ought to be a coherent acronym. "The Long Math..."?

[Monte314]

:laugh: Now that "pronunciation" will be forever stuck in my head.



Thanks. It ought to be a coherent acronym. "The Long Math..."?

OK, I have proven a number theoretic result which I will post as the next problem. It will be called Wufnu's Theorem.

[TheLastMohican]

OK, I have proven a number theoretic result which I will post as the next problem. It will be called Wufnu's Theorem.

"Wufnu's Theorum." It has a zany ring to it. :thumbsup:

[Monte314]

Answers in ():

Easy Questions:

1.) (-5832) What is the smallest value that can be obtained as an Antares Expression?
2.) (59,049) What is the largest value that can be obtained as a Antares Expression?

Hard Questions:

3.) (Yes, in 60 ways) Can 2008 be obtained as an Antares Expression?
4.) Can your birth year be obtained as an Antares Expression? (Monte's can... in 132 diferent ways!)

Very Hard Questions:

5.) (0, in 88,520 ways) What value can be obtained from the largest number of different Antares Expressions?

Here are the most frequently occurring values:
Number, Occurrences
0 , 88520
8 , 69420
12 , 69413
6 , 69101
9 , 66324
4 , 64534
10 , 64108
7 , 63341
5 , 62350
3 , 60861
16 , 59392
2 , 58996
14 , 58995
18 , 58295
15 , 58033
11 , 57225

6.) (1,388) What is the smallest positive value that can be only be obtained from one Antares Expression?
7.) (851) What is the smallest positive value that cannot be obtained as an Antares Expression at all?

[Antares]

Wow. I'm flattered, despite not making an effort to solve any of these. Life's funny, isn't it? Naming a mathematical expression after someone who fail so horrendously at the subject itself.

[Monte314]

Wow. I'm flattered, despite not making an effort to solve any of these. Life's funny, isn't it? Naming a mathematical expression after someone who fail so horrendously at the subject itself.

It is our privilege to commemorate your magnificence in this small way, oh brilliant one.