Thursday, January 07, 2010

NPR Sunday Puzzle (Jan. 3, 2010): It All Adds Up to a New Year

NPR Sunday Puzzle (Jan. 3, 2010): It All Adds Up to a New Year:
Q: Write down the digits from 2 to 7, in order. Add two mathematical symbols to get an expression equaling 2010. What symbols are these?
Yippee! A math puzzle for 2010. The most obvious question is, do you need to get fancy with symbols beyond the standard operations of multiplication, division, addition and subtraction? For example, do you need to use a decimal point, factorials, exponentiation, square roots, etc.? Would Will be so diabolical or would he start us off easy in 2010?

I will say, using just the standard four operations between the digits, you can get 160 different results (4 x 4 x C(5,2) = 160). Of these results, 69 are positive integers. Once you have solved the puzzle for 2010, have fun seeing if you can create any of these results: 1, 623, 1102, 1103, 2291, 4572 or 4573. Also, what's the largest number you can create with just the standard operations?

Edit: The largest number you can form is 2345x6x7 = 98,490. If you study that number you'll see it is 49 times the desired solution of 2010. So just divide by 7 instead of multiplying.
A: 2345 x 6 / 7 = 2010

36 comments:

  1. Here's my standard reminder... don't post the answer or any outright spoilers before the deadline of Thursday at 3pm ET. If you know the answer, click the link and submit it to NPR, but don't give it away here. Thank you.

    ReplyDelete
  2. Like Blaine, I also made up another variant of Will’s puzzle for your enjoyment.

    After solving the on-air puzzle, make an expression equaling 2010 using the digits 2 3 4 5 5 6 7 and only two standard mathematical operators.

    Happy New Year everyone!

    Chuck

    ReplyDelete
  3. Lorenzo, you got the largest. Did you get 2010?

    ReplyDelete
  4. A proportion of Genesis might prove to be an inspiration.

    ReplyDelete
  5. This comment has been removed by a blog administrator.

    ReplyDelete
  6. Lorenzo, it's obvious you got the answer so consider my question rhetorical.

    ReplyDelete
  7. Incidentally, the on-air puzzle was Will's "names in the news" quiz. The group here had pegged most of the names last week, but there were a couple overlooked:
    - Laura Ling and Euna Lee
    - Bo

    ReplyDelete
  8. Blaine, a few comments:
    1. Sorry about the spoiler, but you caught it before too much damage was done.

    2. Elegant analysis to get 160 possible combinations.

    3. I assume you wrote a program to find the 69 combinations leading to positive integers. After the deadline, I would be interested in seeing it.

    4. Of your additional challenges, only 2291 eludes me.

    ReplyDelete
  9. Blaine, I think you may be right though your answer is slightly esoteric :) For everyone else, I may not have made myself clear. So at the risk of repeating myself...

    The rules for my puzzle are exactly the same as the rules for Will’s on-air puzzle: use all of the digits in the order given – 2 3 4 5 5 6 7 – insert two mathematical operators and come up with an expression that equals 2010. The only difference between mine and Will’s is that there are two 5s instead of one.

    Chad

    ReplyDelete
  10. All I can say is that this one came pretty easy to me. Since I was an English major in college, I'm assuming one doesn't have to be a rocket surgeon to do the math on this puzzle.

    ReplyDelete
  11. Blaine, you're right (of course): 69 are positive integers (including 2291).

    Actually, there are more if you allow the first operation to be negation with the minus sign placed before the 2.

    ReplyDelete
  12. Blaine, some final comments:

    For none of the 69 is the first operation division.

    Some (like 2291) require attention to the hierarchy of operations.

    And some, like 45 and 535, involve a brief excursion "outside the box".

    ReplyDelete
  13. I wrote a program for this in VBA for Excel.
    Is that cheating? For me, that's usually the fun I get out of solving many puzzles. I did spend about 20 minutes off and on guessing, then 20 minutes writing the program to solve.
    Thanks, Blaine for a great blog.
    A clue for the correct answer? Call on Blaine for the info.

    ReplyDelete
  14. For those who don't want the extra credit math puzzles, here is Ken's Puzzle #2: A Night at the Opera

    Name a job in an opera company (8 letters long). Change the 5th letter to the letter before it in the alphabet. You get another job in an opera company. What are the two jobs?

    ReplyDelete
  15. Ken, I got it! Quite quickly, I might add. (Sorry to all if that sounds self-aggrandizing.)

    I won't post the answer yet in case others want to play. I've clued it, though. Fun puzzle!

    -- Other Ben

    ReplyDelete
  16. Hey Other Ben,
    Great job, and I like your clues.

    ReplyDelete
  17. Ken, I got it too, but I don't want to shout out the answer to the public.

    ReplyDelete
  18. Ken –

    The two jobs I came up with could apply to any professional stage production – a play, a musical, etc. They’re not specifically related to an opera. I’ll be happy to post them if you’d like to see...

    Chuck

    ReplyDelete
  19. My answer to the two opera company jobs are prompter and promoter.

    The answer to my 2345567 question is 2345 – 5 x 67.

    Chuck

    ReplyDelete
  20. Here are the answers to the numbers I posed:
    23+45-67 = 1
    2x345-67 = 623
    23x45+67 = 1102
    234x5-67 = 1103
    23+4x567 = 2291
    2+3+4567 = 4572
    2x3+4567 = 4573

    ReplyDelete
  21. Here's a link to my solution to the problem. It includes all solutions for years 1 thru 3000 and a link to the computer program I wrote in tiny-c.

    http://primepuzzle.com/tc/npr.html

    ReplyDelete
  22. Depending on whether you think of ( ) as a single mathematical symbol or as two of them, here is an alternate answer. Opinions?

    2/3(45)67 = 2010

    -- Other Ben

    ReplyDelete
  23. Tinyc Tim, some of your answers require rounding but we got some of the same answers. I used a spreadsheet and came up with 69 positive integers. As Lorenzo noted, you can get a few more answers if you use a negation sign in front, like -23+4567

    Ben, I think your definition of () as a single symbol might be considered too "creative", but that does indeed come out to 2010. Is that what you submitted?

    ReplyDelete
  24. Here's a similar challenge with the digits 1 to 9, but with any number of symbols:
    2010 Expressions

    ReplyDelete
  25. Thanks for playing Ken's Puzzle #2: A Night at the Opera. Other Ben proved to be the primo uomo of solvers. His response was very quick ... prompter than I expected. This almost caused him to be a self-promoter.

    ReplyDelete
  26. Blaine -

    My program uses "integer arithmetic" (drops all fractional parts when dividing). I've modified it to display results only when there are no remainders when dividing. If you visit

    http://primepuzzle.com/tc/npr.html

    it will have better output. Thanks for getting back.
     

    ReplyDelete
  27. I used an Excel spreadsheet to generate the various expressions, then a macro to evaluate them:
    2010 Expressions

    ReplyDelete
  28. Blaine, How about 0 to 9?

    0!*1+2345*6/7+8-9

    Submitting the above modification to the puzzle's answer as a Google search returns a nicely parsed

    ((0 !) * 1) + ((2 345 * 6) / 7) + 8 - 9 = 2 010

    ReplyDelete
  29. The new week puzzle is up. Think of a familiar 10-letter hyphenated word that uses all seven letters of the alphabet from "F" to "L" plus three other letters of your choosing. What word is it? It's a word everyone knows, and it's in some dictionaries.
    Will has got to be kidding with this one. It wasn't in ANY dictionary I own.

    ReplyDelete
  30. I wrote a program to generate 1000 scrambled arrangements of the 7 letters, 3 spaces and hyphen in hopes I'd see some pattern that helped me find the answer.

    http://primepuzzle.com/tc/MFILE.TXT

    http://primepuzzle.com/tc/bananag.tc

    So far, I'm still pretty much clueless.
     

    ReplyDelete
  31. Tinyc Tim,
    Check the other thread for more clues on the 1/10/2010 puzzle. There are (7+3)!/3! ways (604,800 ways) to arrange 7 distinct items and 3 similar items. So if you want to brute force all the arrangements, you still have a few more combinations to generate and then review.

    ReplyDelete
  32. Can someone email me the answer becuase i give up and i really want to know lol
    JohnCHS11@aol.com

    ReplyDelete
  33. John, if you mean the 1/10 puzzle, I'll give out the answer after the Thu 3pm ET deadline. Check the other post...

    ReplyDelete