Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 4:26 AM #1

Alex Yeilding
Guest

Friday puzzle -- Team effort 2,3,7, &8

This is a two-part challenge. Hopefully, lots of folks will contribute to the first part.

Express nearly every positive integer from 1 to 30 inclusive using all four of the numbers 2,3,7, &8; any or all of the four basic arithmetic functions indicated by +, -, *, and /; and parentheses for grouping. For instance, 3 can be expressed as (7+8)/3-2. (Note, 2,3,7,&8 are numbers here, not digits, so 23-7-8 is not a solution for 8.)

Part one is to find solutions for 29 of these numbers, and thus to identify the one that can't be expressed in those terms.

Part two is to determine what other operator could be added to allow this last one to be solved.

Suggested process: When you find one or several, put the number(s) in the subject line, and the solutions in the body. I'll check in periodically and post a "yet-to-be-found" list.

Computer-types: Writing a program to solve this may also be fun. If you do so, please just announce your accomplishment without revealing any solutions or the impossible number. Then, we'll call on you if our collective efforts fall short.

I'll kick it off with a couple of solutions.

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 4:28 AM #2

Alex Yeilding
Guest

1,2,3

1 = (3+7)/(2+8)

2 = 3+8-7-2

3 = (8+7)/(3+2)

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 6:55 AM #3

Joel M Kramer
Guest

Re: 5

5=(3x7)-(2x8)

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 7:14 AM #4

Joel M Kramer
Guest

Re: 7, 10, 12, 14, 19, 20, 26

7=3x2-7+8

10=3x8-2x7

12=2x8+3-7

14=8/2+3+7

19=(3x8)/2+7

20=2x8-3+7

26=2x8+3+7

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 7:33 AM #5

Joel M Kramer
Guest

Re: 0, 8, 9, 21, 25, 27, 30

0=8/2-(3-7)

8=8/2-3+7

9=2x7-8+3

21=2x3+7+8

25=2x7+3+8

27=7x3+8-2

30=7x8-3-2

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 8:02 AM #6

Joel M Kramer
Guest

Re: 6, 11, 29

6=2+3-7+8

11=(7+2)/3+8

29=3x8+7-2

What is left.

4,13,15,16,17,18,22,23,24,28

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 8:21 AM #7

Joel M Kramer
Guest

Re: 17

17=3x7-8/2

Numbers that are left:

4,13,15,16,18,22,23,24,28,30

previous wrong 30=7x8-3-2

it is 51=7x8-3-2

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 8:45 AM #8

Joel M Kramer
Guest

Re: 13

13=(3+7)/2+8

Numbers that are left:

4,15,16,18,22,23,24,28,30

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 8:53 AM #9

Joel M Kramer
Guest

Re: 15,24

15=(3-2)x(7+8)

24=(7-3)x(8-2)

Nr. left:

04,16,18,22,23,28,30

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 9:02 AM #10

Joel M Kramer
Guest

Re: 16

16=((7-3)/2)x8

Nr. left:

04,18,22,23,28,30

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 9:12 AM #11

Joel M Kramer
Guest

Re: Friday puzzle -- Team effort 2,3,7, &8

0=8/2-(3-7)

1 = (3+7)/(2+8)

2 = 3+8-7-2

3 = (8+7)/(3+2)

4

5=(3x7)-(2x8)

6=2+3-7+8

7=3x2-7+8

8=8/2-3+7

9=2x7-8+3

10=3x8-2x7

11=(7+2)/3+8

12=2x8+3-7

13=(3+7)/2+8

14=8/2+3+7

15=(3-2)x(7+8)

16=((7-3)/2)x8

17=(3x7)-(8/2)

18

19=(3x8)/2+7

20=2x8-3+7

21=2x3+7+8

22

23

24=(7-3)x(8-2)

25=2x7+3+8

26=2x8+3+7

27=7x3+8-2

28

29=3x8+7-2

30

What is left.

4,18,22,23,28,30

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 9:38 AM #12

Joel M Kramer
Guest

Re: 28

28=(3+7)x2+8

nr. left:

4,18,22,23,30

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 12:25 PM #13

Alex Yeilding
Guest

Re: Friday puzzle -- Team effort 2,3,7, &8

0<>8/2-(3-7) , but the fix is easy, an the question was only for 1-30. Rest look good.

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 12:29 PM #14

Alex Yeilding
Guest

Still looking for 4,18,22,23,30

Joel's knocked this one out of the park. Anyone else before he wraps it up?

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 12:31 PM #15

Alex Yeilding
Guest

Computer solvers

Anyone working on a program for this one? If so, it might be interesting to add to your code a count of distinct solutions for each number (or even better, distinct solutions excluding commutation).

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 1:38 PM #16

Dan Donaldson
Member
Offline
5,264 posts

Registered: 9/17/01

4

4=(7+3+2)-8

- - -

I have been woodworking for about 25 years now, and would consider myself an enthusiastic amateur. My claim to fame (or infamy) has always been fixing stuff. I spent time in the Air Force in the 60's then went to work as an electronic technician for civil service. I finally got my EE, and being fed up with the civil service nonsense, took a job with Ford Motor Company until I went with the split to Visteon last year. I got interested in music and started playing in a bluegrass band. I wanted a new guitar, and that kind of led me into woodworking. I bought some books, a few tools, and built a guitar. Luckily, I didn't know any better than to try that as a first ever project. I still play it when I get a chance, so it must not be too bad if it can live for over 20 years. After a stint in Mexico for my job, I moved into a new house and built a small shop. Since then I have been collecting tools and skills, probably more of the former than the latter ;-). Finding this site has done more for my skills and enjoyment than anything else. I have learned a lot and really enjoy the friendships that I have made here. I have been married to LOML (Lori) for 32 years now and have 6 kids. When they are all gone, I want to get a 1 bedroom house (so there is not room for them to move back in) and at least a 6 car heated/air conditioned garage for a shop. (Dream on...)

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 1:51 PM #17

Dan Donaldson
Member
Offline
5,264 posts

Registered: 9/17/01

18

((7*8)-2))/3=18

- - -

I have been woodworking for about 25 years now, and would consider myself an enthusiastic amateur. My claim to fame (or infamy) has always been fixing stuff. I spent time in the Air Force in the 60's then went to work as an electronic technician for civil service. I finally got my EE, and being fed up with the civil service nonsense, took a job with Ford Motor Company until I went with the split to Visteon last year. I got interested in music and started playing in a bluegrass band. I wanted a new guitar, and that kind of led me into woodworking. I bought some books, a few tools, and built a guitar. Luckily, I didn't know any better than to try that as a first ever project. I still play it when I get a chance, so it must not be too bad if it can live for over 20 years. After a stint in Mexico for my job, I moved into a new house and built a small shop. Since then I have been collecting tools and skills, probably more of the former than the latter ;-). Finding this site has done more for my skills and enjoyment than anything else. I have learned a lot and really enjoy the friendships that I have made here. I have been married to LOML (Lori) for 32 years now and have 6 kids. When they are all gone, I want to get a 1 bedroom house (so there is not room for them to move back in) and at least a 6 car heated/air conditioned garage for a shop. (Dream on...)

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 5:30 PM #18

Alex Yeilding
Guest

And then there were three: 22, 23, and 30

Two of which are solvable.

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 7:22 PM #19

Joel M Kramer
Guest

Re: Friday puzzle -- Team effort 2,3,7, &8

0<>8/2-(3-7) , but the fix is easy, an the question was only for 1-30.

0 = (8/2)+(+3-7) sorry reversed the neg. sign.

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 8:06 PM #20

Joel M Kramer
Guest

Re: And then there were three: 22, and 30

23=(2+8)x3-7

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 8:23 PM #21

Alex Yeilding
Guest

23 solved, one of 22 or 30 is possible

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 8:46 PM #22

Joel M Kramer
Guest

Re: And then there were one: 22

0 = 8/2+(3-7)

1 = (3+7)/(2+8) or (7-3)/(8/2)

2 = 3+8-7-2

3 = (8+7)/(3+2)

4 = 7+3+2-8

5 = (3x7)-(2x8)

6 = 2+3-7+8

7 = 3x2-7+8

8 = 8/2-3+7

9 = 2x7-8+3

10 = 3x8-2x7

11 = (7+2)/3+8

12 = 2x8+3-7

13 = (3+7)/2+8

14 = 8/2+3+7 or ((8-2)x7)/3

15 = (3-2)x(7+8) or (3+8)x2-7

16 = ((7-3)/2)x8

17 = (3x7)-(8/2)

18 = ((7*8)-2)/3

19 = (3x8)/2+7 or 2x7+8-3

20 = 2x8-3+7 or 8+7+3+2

21 = 2x3+7+8

22 =

23 = (2+8)x3-7

24 = (7-3)x(8-2)

25 = 2x7+3+8 or (7x8)/2-3 or 3x7+8/2

26 = 2x8+3+7

27 = 7x3+8-2 or (7+8)x2-3

28 = (3+7)x2+8

29 = 3x8+7-2 or (3+8)x2+7

30 = (7-3)x8-2

31 = (8x7)/2+3

Re: Friday puzzle -- Team effort 2,3,7, &8

10/16/09, 9:11 PM #23

Alex Yeilding
Guest

Now on to part 2

22 cannot be expressed with 2,3,7,&8 and the four most basic arithmetic operations. What one additional operator can be added to get a solution for 22, and what is that solution? (There may be more than one answer.)

Re: Friday puzzle -- Team effort 2,3,7, &8

10/17/09, 5:55 PM #24

Larry Barrett
Member
Offline
1,070 posts

Registered: 3/11/04

Re: Now on to part 2

How about ((8!/7!)x3)-2.

Re: Friday puzzle -- Team effort 2,3,7, &8

10/17/09, 7:19 PM #25

Dan Donaldson
Member
Offline
5,264 posts

Registered: 9/17/01

Beautimous ;-)

That is one operation that I didn't think of.

- - -

I have been woodworking for about 25 years now, and would consider myself an enthusiastic amateur. My claim to fame (or infamy) has always been fixing stuff. I spent time in the Air Force in the 60's then went to work as an electronic technician for civil service. I finally got my EE, and being fed up with the civil service nonsense, took a job with Ford Motor Company until I went with the split to Visteon last year. I got interested in music and started playing in a bluegrass band. I wanted a new guitar, and that kind of led me into woodworking. I bought some books, a few tools, and built a guitar. Luckily, I didn't know any better than to try that as a first ever project. I still play it when I get a chance, so it must not be too bad if it can live for over 20 years. After a stint in Mexico for my job, I moved into a new house and built a small shop. Since then I have been collecting tools and skills, probably more of the former than the latter ;-). Finding this site has done more for my skills and enjoyment than anything else. I have learned a lot and really enjoy the friendships that I have made here. I have been married to LOML (Lori) for 32 years now and have 6 kids. When they are all gone, I want to get a 1 bedroom house (so there is not room for them to move back in) and at least a 6 car heated/air conditioned garage for a shop. (Dream on...)

Friday puzzle -- Team effort 2,3,7, &8

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