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Ease Into The New Year

Posted by Karl Sharman on January 1, 2011 – 9:47 am

1. Karl has half the Jelly Babies that Chris has. Wiz has 3 times as many as Karl. Together they have 72. How many Jelly Babies does each one have?

2. I am 14 inches taller than Al Gelman. The difference between myself and Nordtorp is two inches less than between Nordtorp and Al Gelman. At 6′6″, I am the tallest. How tall are Nordtorp and Al Gelman?

3. The Puzzlania Rugby Team had just won the championship game and the players wanted to congratulate each other. They began shaking hands, but each player only shook hands with every other player just once. There are, of course, only 15 players on a rugby team. How many times did the players shake hands?

4. A jar has 4 amoebas in it to start. Amoebas split their cells in two (therefore doubling in size) once every minute. The jar will be completely filled in 10 minutes. How long would it take to fill the same sized jar if had 8 amoebas in it to start?

5. At the Annual Puzzlania Golf Tournament, 100 golfers enter. I didn’t bother this year, as I have won the last 10 years in a row, and felt it was time to let the less skilful players have a chance. 100 golfers start the contest. If a player loses a match, he is immediately eliminated from the contest. How many matches will be played to determine the winner?

6. Find a simple method of solving:
6751X + 3249Y = 26751
3249X + 6751Y = 23249

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14 Responds so far- Add one»

  1. 1. Nathan Said:

    1. Karl has 12, Chris has 24, and Wiz has 36.

  2. 2. Nathan Said:

    2. Al is 5′4″, Northrop is 6′0″
    3. 120
    4. 9 min
    5. 99 matches
    6. x=3,y=2

  3. 3. Karl Sharman Said:

    Nathan’s has been working hard. Well done! All right…. well almost all. You got more than 1 right, but less than 6 right.
    Hope this helps….

  4. 4. Wizard of Oz Said:

    I think Nathan only has has 3. wrong, should be 105 handshakes.

  5. 5. Chris Said:

    3. Each player on one team shook hands with each plsyer on the other tesm. So that’s 15 handshakes for each plsyer. So that’s 15*15 = 225 handhakes altogether. But you could say that one handshake counts as two people shaking hands and so get 450. (What have I missed????)

    Otherwise I concur with Nathan’s results.

  6. 6. cazayoux Said:

    6. (showing my work)

    I added the two formulae

    10000x + 10000y = 50000
    x + y = 5

    Then I subtracted the two formulae.
    I didn’t need the exact number since i know the result for each would be the same.

    Ax – Ay = A
    x – y = 1

    Easy from the two derived
    y = x – 1
    x + x – 1 = 5
    2x = 4
    x = 2
    y = 3

  7. 7. Al Gelman Said:

    You were right the first time, Chris. The question was to count handshakes not people. so 15*15=225 is correct.

  8. 8. Chris Said:

    Hi cazayoux. Very nice, well spotted.

    You goofed your write-up when you posted.
    Starting with x+y = 5 and x-y = 1. 2nd => y = x-1. Subst into 1st =>
    x + x-1 = 5 => 2x = 6 (not 4), so x = 3 and y = 2.

    Personally, I would have added the two equations to get 2x = 6 directly. Then I would have subtracted them to get 2y = 4 directly.

  9. 9. Karl Sharman Said:

    Well, it appears that these were all too easy. That is, all except the hand-shakes.
    Only one person has it right for 1 team of 15. So what is it for 2 teams of 15?

  10. 10. Chris Said:

    Oops! I misread the question. I thought all the players on one team were shaking hand with all the players on the other team. I hadn’t realised that the Puzzlanians had formed a self-admiration society.

    So for the Puzzlanians, the first shakes hands with the other 14, the second shakes hands with the other 13, …, the 14th shakes hands with the 15th. So that’s (14/2)(1+14) = 105, thank you Karl (Gauss ;) ), as Nathan said.

    Follow up challenge. Double the 105 if the teams won’t touch each other, and (29/2)(1+29) = 435 if all 30 shake hands with each other.

  11. 11. Karl Sharman Said:

    Yup all questions now answered correctly.

    Explanations follow:-

    1. If Chris has twice as many Jelly Babies as Karl and Wiz has 3 times as many as Karl, then the total of 72 must be divided by 6. 72÷6=12. Therefore: Karl has 12 Jelly Babies, Chris has 24 Jelly Babies, and Wiz has 36 Jelly Babies

    2. If I am 6′6″ tall, Al Gelman must be 5′4″ tall, and Nortorp must be 6′ tall because he is 6″ shorter than me and 8″ taller than Al Gelman

    3. The first player only shakes hands with 14 other players, the second player only shakes hands with 13 other players (he already shook hands with the first player!), the third player only shakes hands with 12 players ( he already shook hands with the first player and second player ), and so on, until the last player who only has one possibility left. Therefore: 14+13+12+11+10+9+8+7+6+5+4+3+2+1=105

    4. 9 minutes. Working backwards from how many amoeba you have at ten minutes might make it easier. But 8 amoeba is the number after the first division, which takes place after 1 minute – 10-1 = 8 minutes. Lets see if anyone reads this explanation!

    5. If there is to be only one winner, than there are 99 losers, therefore there were 99 matches.
    You can also do this with math: in round 1 there would be 50 matches to produce 50 winners, round 2 would have 25 matches to produce 25 winners, round 3 would have 12 matches to produce 12 winners ( one person of the 25 winners would have to wait until later to play again ), round 4 would have 6 matches to produce 6 winners, round 5 would have 3 matches to produce 3 winners, round 6 would have 2 matches ( the player left out before would now play to make it an even field ) to produce 2 winners, these 2 would play for the championship. So: 50+25+12+6+3+2+1=99

    6. Add the equation to give: 10000X + 10000Y = 50000
    Divide by 10000 to give: X + Y = 5

    Subtract the equation to give: 3502X – 3502Y = 3502
    Divide by 3502 to give: X – Y = 1

    Therefore: X = 3 and Y = 2, although I think Chris’s post 8 is a better answer…

  12. 12. anonymous Said:

    i read the explanation and saw that 10-1 isn’t 8 /gg

  13. 13. Karl Sharman Said:

    Thanks Anonymous!

  14. 14. Shannon Said:

    1. Chris= x Karl= (1/2)x Wiz= (3/2)x
    x + (1/2)x + (3/2)x = x + 2x = 3x = 72
    x = 24 (1/2)x = 12 (3/2)x = 36

    2. Me= 6′6″ Al= 6′6″-14″= 5′4″ Nordtorp= 5′4″+2″= 5′6″

    3. 15 (players shaking) x 14 (being shook) = 210 handshakes

    4. 4 x2×2…(ten times)= 4096 8 x2×2…(nine times)= 4096

    5. Sorry, I on’t know enough about golf to know how a competition works and can’t answer the question.

    6. Add formulas: Subtract formulas:
    10000X + 10000Y = 50000 3502X – 3502Y = 3502
    Divide by 10000: Divide by 3502:
    X + Y = 5 X – Y = 1
    X = 3 3 + 2 = 5
    Y = 2 3 – 2 = 1

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