Math: Antares Expressions

**Monte314** · 09-03-2008, 06:58 PM

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** · 09-03-2008, 08:44 PM

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

( Show Spoiler )

...And the "hard" questions:

( Show Spoiler )

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** · 09-03-2008, 08:48 PM

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** · 09-03-2008, 08:49 PM

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	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** · 09-04-2008, 02:23 AM

( Show Spoiler )

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:

( Show Spoiler )

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:

( Show Spoiler )

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...

( Show Spoiler )

**Monte314** · 09-04-2008, 06:19 AM

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** · 09-05-2008, 03:13 PM

Wow, she already has her own expression. That is quite an accomplishment.
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

**TheLastMohican** · 09-05-2008, 03:51 PM

	Originally Posted by Tenacious B To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Wow, she already has her own expression. That is quite an accomplishment. To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

Indeed. I nominate Wufnu for the next one.

**Wufnu** · 09-06-2008, 03:14 AM

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** · 09-06-2008, 05:30 AM

Zzzzzz huh what?

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

Don't make me think too much.

**ssrprotege** · 09-06-2008, 05:43 AM

	Originally Posted by TheLastMohican To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Indeed. I nominate Wufnu for the next one.

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

**TheLastMohican** · 09-06-2008, 01:05 PM

	Originally Posted by Wufnu To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	Mine sounds like a nerd sneezing.

To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
Now that "pronunciation" will be forever stuck in my head.

	Originally Posted by ssrprotege To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	I nominate TLM the next. TLM Expression doesn't sound too bad.

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

**Monte314** · 09-06-2008, 03:07 PM

	Originally Posted by TheLastMohican To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts. 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** · 09-06-2008, 03:52 PM

	Originally Posted by Monte314 To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	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.
To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.

**Monte314** · 09-07-2008, 04:07 PM

( Show Spoiler )

**Antares** · 09-10-2008, 02:01 AM

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** · 09-14-2008, 06:17 PM

	Originally Posted by Antares To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
	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.

09-03-2008, 06:58 PM	#1
Monte314 Core Member [407%] MBTI: INTJ Join Date: Apr 2008 Posts: 16,286	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: 2-42-9+2 = (((2-4)2)-9)+2 = ((-22)-9)+2 = (-4-9)+2 = -13+2 = -11 (We can also obtain it, for example, as 6-752-1) There are 939393939 = 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? Last edited by Monte314; 09-03-2008 at 08:25 PM.*
Monte314 is offline post a comment

09-03-2008, 08:44 PM	#2
TheLastMohican Core Member [187%] MBTI: ENTJ Join Date: Feb 2008 Posts: 7,498	Okay, I can handle the "easy" questions (love the hints!): ( Show Spoiler ) 1. (1-9)999 = -5,832 2. 99999 = 59,049 ...And the "hard" questions: ( Show Spoiler ) 3. 9874-8 = 2008 4. 995-75 = 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. Last edited by TheLastMohican; 09-03-2008 at 09:16 PM.*
TheLastMohican is offline post a comment

09-03-2008, 08:48 PM	#3
Monte314 Core Member [407%] MBTI: INTJ Join Date: Apr 2008 Posts: 16,286	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!
Monte314 is offline post a comment

09-04-2008, 02:23 AM	#5
Wufnu Member [03%] MBTI: INTj Join Date: Aug 2008 Posts: 144	( Show Spoiler ) 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: ( Show Spoiler ) 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 123=213, ie: same expression). 3 will not work (51 = 9+83 = 95+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: ( Show Spoiler ) We're now looking for a number larger than 738 = 999+9. 816 = (9+8)68 seems to fit this bill. Following the same logic, we need a number larger than 9999+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 * ([92(982-1)]-1) certainly works but I doubt it's the lowest. Wufnu added to this post, 64 minutes and 44 seconds later... ( Show Spoiler ) 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 = 180 945+1-1 = 180 945+2-1 = 181 945+1+1 = 182 etc until 945+1+8 = 189 945+2+8 = 190 945+2+9 = 191 etc until 945+9+9 =198 switching to base 9 number system: 22 9 + 1 = 199 ((54)+2)9+1 = 199 ((54)+2)9+9 = 207 ((54)+3)9+1 = 208 = 239 + 1 etc until ((54)+5)9 +1 = 226 = 559+2-1 559+1+1 = 227 this continues until you have (99+9) * 9 + 9= 819 = 9 * 91 820 = (94 + 5) 5 4 For 821 to exist within the rules, no number 821 +/- 9 must be found for any four numbers. 999 = 729 = 819 919 = 819 and 929 = 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 = 6137 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.
Wufnu is offline post a comment

09-04-2008, 06:19 AM	#6
Monte314 Core Member [407%] MBTI: INTJ Join Date: Apr 2008 Posts: 16,286	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. Last edited by Monte314; 09-04-2008 at 02:20 PM.
Monte314 is offline post a comment

09-05-2008, 03:13 PM	#7
Tenacious B Member [10%] MBTI: iNTj Join Date: Mar 2008 Posts: 421	Wow, she already has her own expression. That is quite an accomplishment. To view links or images in this forum your post count must be 2 or greater. You currently have 0 posts.
Tenacious B is offline post a comment

09-06-2008, 03:14 AM	#9
Wufnu Member [03%] MBTI: INTj Join Date: Aug 2008 Posts: 144	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.
Wufnu is offline post a comment

09-06-2008, 05:30 AM	#10
SiMey Member [06%] MBTI: INTJ Join Date: Jun 2008 Posts: 261	Zzzzzz huh what? I used to like maths but gave it up years ago. Don't make me think too much.
SiMey is offline post a comment

09-07-2008, 04:07 PM	#15
Monte314 Core Member [407%] MBTI: INTJ Join Date: Apr 2008 Posts: 16,286	( Show Spoiler ) 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?
Monte314 is offline post a comment

09-10-2008, 02:01 AM	#16
Antares Core Member [170%] MBTI: INTJ Join Date: Dec 2007 Posts: 6,808	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.
Antares is offline post a comment