Dividing Pizza with a Clock

Source: Alok Goyal Puzzle Page ( http://alokgoyal1971.com/ ) . Alok is ex-IIT Delhi, Partner at Helion VC




Problem:

Part I (Easy): Using a clock, divide a pizza among 12 people

Part II (Difficult): Using a clock, divide a pizza among 11 people?




Solution:

Highlight the part between the * symbols for the answer.
* Part I (12 people): cut along the radii pointing at the 12 hour marks - they are evenly spaced at 360/12 = 30 degrees, giving 12 equal slices.

Part II (11 people): use the 11 moments in 12 hours when the hour and minute hands exactly overlap (12:00, ~1:05:27, ~2:10:54, ..., ~10:54:33). The minute hand gains 330 degrees per hour on the hour hand, so successive overlaps are 12/11 hours apart, and the overlap direction advances by exactly 360/11 degrees each time. Those 11 radii are therefore equally spaced - cutting along them divides the pizza into 11 equal slices.

Bonus extension from the comments: minute and second hands overlap 59 times an hour - the same trick gives 59 equal slices (and hour/minute/second overlaps give other counts).

Solution by an Anonymous commenter (Mar 2015) from the comments. *

Comments

  1. Unknown9:36 PM, March 06, 2015

    a) use the 12 marks corresponding to 1'o clock, 2'o clock, ...., 12'o clock
    b) use the 11 distinct and symmetric positions of the clock where the minute hand and hour hand overlap.

    thought of a couple of further extensions:
    c) (easy once previous parts are done) make 59 equal slices.
    d) (mayb hard even after previous parts are done) make 43140 equal slices (not sure if the pizza would be edible by the time u finish this arduous task :D)

    ReplyDelete
    Replies
    1. Unknown11:17 PM, September 09, 2015

      for c) i draw the analogy from part b, since hour and minute hand overlap 11 times, i think minute and second hand will overlap 59 times in an hour, so we can devide in 59 parts.

      Delete
    2. Unknown11:19 PM, September 09, 2015

      for c) i draw the analogy from part b, since hour and minute hand overlap 11 times, i think minute and second hand will overlap 59 times in an hour, so we can devide in 59 parts.

      Delete
  2. Blue sword10:01 PM, April 21, 2015

    Omkar's b) was good
    Another method ( not that easy though )
    We will use the Hours hand to mark the cut and the minutes hand for keeping the count. The total 12 hours consists of 12 * 60 minutes = 720 minutes. If we divide the 720 minutes by 11 we get 65.45 If the hand is currently at 12'o' clock, we keep on extending the minutes hand by 65 minutes ( approx )

    ReplyDelete
  3. Unknown3:37 PM, December 13, 2015

    (d) was really Nice!
    We make the count using minute hand and cut using hour hand. So we get 12*60 = 720min. But first cut can be made at start of each hour,so making it 719. Multiplying it with number of different second hand moves,we get 719*60 =43140 different slices.

    ReplyDelete
  4. SHIVANSHU CHAUHAN10:00 AM, March 22, 2016

    Start when Hour, minute and second overlap on 12'O Clock. Then after 5 minute and 57 seconds, meaning when the minute hand is somewhat ahead of 1 and second hand is on 57 then take the angle b/w hour hand and minute hand. Which is approximately (360/11) degree.

    ReplyDelete

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