1. Subscribervenda
    Dave
    S.Yorks.England
    Joined
    18 Apr '10
    Moves
    72896
    12 Jul '21 14:00
    This weeks puzzle is(yet another) that i don't know how to do without resorting to trial and error:-
    700 stalls were in a row at a festival and each saw one more visitor than the previous.Each stall wrote down it's attendance and the 700 cards were passed to the auditor,who lost one.The auditor totalled the remaining cards as 700,000.
    What number went missing
  2. SubscriberAnitya
    Shoshin
    under a fig tree
    Joined
    13 Sep '20
    Moves
    21408
    12 Jul '21 23:54
  3. Joined
    08 Dec '06
    Moves
    26924
    13 Jul '21 06:192 edits

    Removed by poster

  4. Joined
    08 Dec '06
    Moves
    26924
    13 Jul '21 06:231 edit
    I recognized it as an arithmetic progression (AP) with a common difference of 1.
    A term in an AP has the for a + (n-1)d where a is the first term, n is the number of terms and d is the common difference.
    So the term is a+699.
    The sum of an AP is S(n) = (n/2){2a+(n-1)d}. So the sum is S(700) = (700/2){2a+699}. This simplifies to S(700)= 700a+244650.

    Call the missing card m. The mth term would be a+(m-1).
    The auditor found the total 700a+244650 -[a+(m-1)]=700000.
    700a+244650 -[a+(m-1)]=700000
    699a+244651-m=700000
    699a-m=455349
    Make a the subect of the formula
    a=(455349+m)/699
    The numbers a and m are counting numbers.
    Divide 455349 by 699 to get a number q and a remainder r. So
    q +r/699 +m/699 would be a.
    699=r+m and a=q+1. So you get m by finding 699-r.
    q=651 and r =300
    m=699-300=399
    a=651+1=652
    Card number 399 is missing.

    Checking the answer
    The sum is S(700)=700(352)+244650=701050
    The mth term is 652+398=1050
    The auditor's total is 701050-1050=700000

    So card number 399 is missing.
  5. Subscribervenda
    Dave
    S.Yorks.England
    Joined
    18 Apr '10
    Moves
    72896
    13 Jul '21 08:16
    @anitya said
    [hidden]350[/hidden]
    Incorrect I'm afraid
  6. Subscribervenda
    Dave
    S.Yorks.England
    Joined
    18 Apr '10
    Moves
    72896
    13 Jul '21 08:19
    @damionhonegan
    Correct and you've even taken it a stage further than the question which only gives the answer as the number of stall visitors missing!(1050)
    Well done
  7. SubscriberAnitya
    Shoshin
    under a fig tree
    Joined
    13 Sep '20
    Moves
    21408
    14 Jul '21 00:07
    @venda said
    Incorrect I'm afraid
    ah well
  8. Joined
    12 Jul '08
    Moves
    13787
    14 Jul '21 14:15
    @damionhonegan

    How did you calculate q?
  9. Joined
    08 Dec '06
    Moves
    26924
    16 Jul '21 01:45
    @Eladar

    455349÷699= 651 with a remainder of 300. I called the remainder r. I made q be the quotient without the remainder.
  10. Joined
    12 Jul '08
    Moves
    13787
    16 Jul '21 01:561 edit
    @damionhonegan

    Ok, got it now. I was missing that d=1, therefore simplifies out in both the nth term and adding in the mth term.

    Thanks
  11. Subscriberrookie54
    free tazer tickles..
    wildly content...
    Joined
    09 Mar '08
    Moves
    172385
    23 Jul '21 16:31
    i don't have the brainpower


    https://fivethirtyeight.com/features/can-you-hop-across-the-chessboard/
Back to Top

Cookies help us deliver our Services. By using our Services or clicking I agree, you agree to our use of cookies. Learn More.I Agree