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
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.
@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
@Eladar
455349÷699= 651 with a remainder of 300. I called the remainder r. I made q be the quotient without the remainder.
@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
@damionhonegan saidI do look every week but most of them are not very good so I don't bother posting them.
Do you have any more newspaper puzzles?
This weeks is not very challenging but here goes:-
I have a square.I increase the length of each side by 14 metres and it's area grows by 700 square metres(there's always a 700 element to them!!)What is the area of my new larger square