The Age Game (math challenge)

Discussion in 'Community Discussion' started by Doctor Q, Jan 23, 2009.

  1. Doctor Q Administrator

    Doctor Q

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    #1
    Try the Age Game:
    Step 1: Start with your age in years.
    Step 2: If it's an even number, divide it by two. Otherwise, triple it and add one.
    Step 3: Repeat Step 2 until the answer is 1.​

    Examples: Suppose you are 13 years old. You get to 1 in nine steps:
    13 is odd. 13 x 3 + 1 = 40
    40 is even. 40 / 2 = 20
    20 is even. 20 / 2 = 10
    10 is even. 10 / 2 = 5
    5 is odd. 5 x 3 + 1 = 16
    16 is even. 16 / 2 = 8
    8 is even. 8 / 2 = 4
    4 is even. 4 / 2 = 2
    2 is even. 2 / 1 = 1​

    Suppose you are 70 years old. You get to 1 in 14 steps:
    70 is even. 70 / 2 = 35
    35 is odd. 35 x 3 + 1 = 106
    106 is even. 106 / 2 = 53
    53 is odd. 53 x 3 + 1 = 160
    160 is even. 160 / 2 = 80
    80 is even. 80 / 2 = 40
    40 is even. 40 / 2 = 20
    20 is even. 20 / 2 = 10
    10 is even. 10 / 2 = 5
    5 is odd. 5 x 3 + 1 = 16
    16 is even. 16 / 2 = 8
    8 is even. 8 / 2 = 4
    4 is even. 4 / 2 = 2
    2 is even. 2 / 1 = 1​
    Can you follow the steps for your age without making a mistake? (You may use a calculator.) My apologies to 27-year-olds, but don't give up!

    As an extra incentive, I'll consider you to be a winner if you encounter the number 40 along the way, as in the examples above. If you don't encounter 40, you aren't necessarily a loser; it's merely inconclusive. Are you a confirmed winner of the Age Game?

    Examples:
    17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 (winner)

    24 -> 12 -> 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 (inconclusive)​
    Other questions, just for fun:
    1. Why are powers of two special?

    2. Are there any ages for which the number of steps equals the age? (The number of steps is the number of arrows.) Hint: Test kids' ages.

    3. Are there any ages for which you don't eventually get to 1?

    4. What if we tried the Age Game on the ages of stars in nanoseconds instead of the ages of humans in years. Would we always get to 1?​
     
  2. bassproguy07 macrumors 6502a

    bassproguy07

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    #2
    haha I will have to try this....i suck at math so I will let you know when I figure it out!

    EDIT:20->10->5->16->8->4->2->1 7 steps! correct me if I did this wrong
     
  3. Demosthenes X macrumors 68000

    Demosthenes X

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    Oct 21, 2008
    #3
    They're special because they'll always divide down to one without returning an odd number.

    Also, if you're 5 years old, it takes five steps:
    5 -> 16 -> 8 -> 4 -> 2 -> 1

    I'm pretty sure there's no number that won't eventually reduce to one... the problem is set up so that you can't get into an infinite loop. It'll never return the same number twice.

    And yes, the results would be the same regardless of what measure of time you used.
     
  4. WildCowboy Administrator/Editor

    WildCowboy

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    #4
    Just add a few steps onto the front of your "70 year old" example and you get my winning path:

    30 -> 15 -> 46 -> 23 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
     
  5. atszyman macrumors 68020

    atszyman

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    #5
    Bah, it's easy 32->16->8->4->2->1... benefit of being 2^5 years old...

    1) answered above
    2) is it 5->16->8->4->2->1
    3) possibly, but I don't think people can live that long?

    Edit:The only way to get stuck in an infinite loop is to hit 3x+1=2^n*x for 3x+1=2x the answer is -1... for 3x+1=4x the answer is 1 and for 3x+1=8x the answer is 1/5 and everything after that is non integer so I'm going to go with no you'll always reach one eventually unless you use fractions in your age.


    4) maybe? Edit: see #3 above.
     
  6. siurpeeman macrumors 603

    siurpeeman

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    #6
    29->88->44->22->11->34->17->52->26->13->40->20->10->5->16->8->4->2->1->sadness.
     
  7. EricNau Moderator emeritus

    EricNau

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    Apr 27, 2005
    Location:
    San Francisco, CA
    #7
    18 > 9 > 28 > 14 > 7 > 22 > 11 > 34 > 17 > 52 > 26 > 13 > 40 > 20 > 10 > 5 > 16 > 8 > 4 > 2 > 1

    Shall I start with 3E+25 and let you know? :p
     
  8. tkidBOSTON macrumors 6502a

    tkidBOSTON

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    #8
    At age 26, Siurpeeman has me covered in his calc.

    Maybe I'll revisit this in two months when I hit the fun age of 27. ;)
     
  9. joepunk macrumors 68030

    joepunk

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    #9
    I accept your apology :p

    will post results asap
     
  10. tobefirst macrumors 68040

    tobefirst

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    Jan 24, 2005
    Location:
    St. Louis, MO
    #10
    I'll represent the 28 year olds. :)

    28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

    Just 18 simple steps!


    Doctor Q, this reminds me of my family's house rules for the game "Sorry!" For the evens, we divide by two, for the odds, we triple them, and for the Sorry! cards, everyone else gets to put a guy on the board and the person drawing the card loses a guy. There are some other minor rules (like the 7 card, which becomes a 21, can be split among however many guys you have), but those are the general rules. It's a lot of fun.
     
  11. Doctor Q thread starter Administrator

    Doctor Q

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    #11
    Your game play was correct, bassproguy07.

    You are correct, Demosthenes X, about powers of two. They never go through the multiplication step.

    atszyman, you are correct that age 5 takes 5 steps. That's the only human age with that property.

    A tip: While playing, notice if you come to an age that's already shown in someone else's results. You can copy the rest of the sequence from them. Example:
    22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 [stealing from bassproguy07] -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 (15 steps)​


    I've been testing various large numbers, to see how many steps they take, and encountered a surprising result:

    These five consecutive large integers all take 52 steps! They take a couple of paths to get to 994, then synchronize from there.
    8811 -> 26434 -> 13217 -> 39652 -> 19826 -> 9913 -> 29740 -> 14870 -> 7435 -> 22306 -> 11153 -> 33460 -> 16730 -> 8365 -> 25096 -> 12548 -> 6274 -> 3137 -> 9412 -> 4706 -> 2353 -> 7060 -> 3530 -> 1765 -> 5296 -> 2648 -> 1324 -> 662 -> 331 -> 994 -> 497 -> 1492 -> 746 -> 373 -> 1120 -> 560 -> 280 -> 140 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

    8812 -> 4406 -> 2203 -> 6610 -> 3305 -> 9916 -> 4958 -> 2479 -> 7438 -> 3719 -> 11158 -> 5579 -> 16738 -> 8369 -> 25108 -> 12554 -> 6277 -> 18832 -> 9416 -> 4708 -> 2354 -> 1177 -> 3532 -> 1766 -> 883 -> 2650 -> 1325 -> 3976 -> 1988 -> 994 -> 497 -> 1492 -> 746 -> 373 -> 1120 -> 560 -> 280 -> 140 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

    8813 -> 26440 -> 13220 -> 6610 -> 3305 -> 9916 -> 4958 -> 2479 -> 7438 -> 3719 -> 11158 -> 5579 -> 16738 -> 8369 -> 25108 -> 12554 -> 6277 -> 18832 -> 9416 -> 4708 -> 2354 -> 1177 -> 3532 -> 1766 -> 883 -> 2650 -> 1325 -> 3976 -> 1988 -> 994 -> 497 -> 1492 -> 746 -> 373 -> 1120 -> 560 -> 280 -> 140 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

    8814 -> 4407 -> 13222 -> 6611 -> 19834 -> 9917 -> 29752 -> 14876 -> 7438 -> 3719 -> 11158 -> 5579 -> 16738 -> 8369 -> 25108 -> 12554 -> 6277 -> 18832 -> 9416 -> 4708 -> 2354 -> 1177 -> 3532 -> 1766 -> 883 -> 2650 -> 1325 -> 3976 -> 1988 -> 994 -> 497 -> 1492 -> 746 -> 373 -> 1120 -> 560 -> 280 -> 140 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1

    8815 -> 26446 -> 13223 -> 39670 -> 19835 -> 59506 -> 29753 -> 89260 -> 44630 -> 22315 -> 66946 -> 33473 -> 100420 -> 50210 -> 25105 -> 75316 -> 37658 -> 18829 -> 56488 -> 28244 -> 14122 -> 7061 -> 21184 -> 10592 -> 5296 -> 2648 -> 1324 -> 662 -> 331 -> 994 -> 497 -> 1492 -> 746 -> 373 -> 1120 -> 560 -> 280 -> 140 -> 70 -> 35 -> 106 -> 53 -> 160 -> 80 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1​
     
  12. tkidBOSTON macrumors 6502a

    tkidBOSTON

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    #12
    I felt cheated for having the 26 done for me, so I went and did the 27...

    Code:
    27
    82
    41
    124
    62
    31
    94
    47
    142
    71
    214
    107
    322
    161
    484
    242
    121
    364
    182
    91
    274
    137
    412
    206
    103
    310
    155
    466
    233
    700
    350
    175
    526
    263
    790
    395
    1186
    593
    1780
    890
    445
    1336
    668
    334
    167
    502
    251
    754
    377
    1132
    566
    283
    850
    425
    1276
    638
    319
    958
    479
    1438
    719
    2158
    1079
    3238
    1619
    4858
    2429
    7288
    3644
    1822
    911
    2734
    1367
    4102
    2051
    6154
    3077
    9232
    4616
    2308
    1154
    577
    1732
    866
    433
    1300
    650
    325
    976
    488
    244
    122
    61
    184
    92
    46
    23
    70
    35
    106
    53
    160
    80
    40
    20
    10
    5
    16
    8
    4
    2
    1
     
  13. atszyman macrumors 68020

    atszyman

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    Location:
    The Dallas 'burbs
    #13
    All integers from 1 to 3824 will reduce to 1 in less than 237 steps the lone number taking all 237 steps being 3711

    For those who want to take the math out of it use the following formula in Excell. Past it into A2-Awhatever and put your age in A1.

    =IF(A1=1,A1,IF(MOD(A1,2)=0,A1/2,A1*3+1))

    It will start to repeat 1s once you hit the end. The number on the A cell with the final 2 is the number of steps it took to complete.
     
  14. swiftaw macrumors 603

    swiftaw

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    Location:
    Omaha, NE, USA
    #14
    31->94->47->142->71->214->107->322->161->484->242->121->364->182->91->274->137->412->206->103->310->155->466->233->700->350->175->526->263->790->395->1186->593->1780->890->445->1336->668->334->167->502->251->754->377->1132->566->283->850->425->1276->->319->958->479->1438->719->2158->1079->3238->1619->4858->2429->7288->3644->1822->911->2734->1367->4102->2051->6154->3077->9232->4616->2308->1154->577->1732->866->433->1300->650->325->976->488->244->122->61->184->92->46->23->70->35->106->53->160->80->40->20->10->5->16->8->4->2->1

    106 steps
     
  15. OutThere macrumors 603

    OutThere

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    NYC
    #15
    20->10->5->16->8->4->2->1

    It's good to be young :p
     
  16. tkidBOSTON macrumors 6502a

    tkidBOSTON

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    #16
    Curses!
    I was trying to find an excel formula for that!
     
  17. rdowns macrumors Penryn

    rdowns

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    Jul 11, 2003
    #17
    I think I did this correctly.

    46-23-70-35-106-53-160-80-40-20-10-5-16--8-4-2-1
     
  18. Queso macrumors G4

    Joined:
    Mar 4, 2006
    #18
    Hell, I've got a lot of steps. 22 in total :eek:

    36 > 18 > 9 > 28 > 14 > 7 > 22 > 11 > 34 > 17 > 52 > 26 > 13 > 40 > 20 > 10 > 5 > 16 > 8 > 4 > 2 > 1.
     
  19. atszyman macrumors 68020

    atszyman

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    #19
    Why curses? I just saved you the work?

    5 is the only number between 1 and 3824 that finishes in the same number of steps as it's value.

    I've seen a lot of consecutive integers end up at the same point 54 and 55, 62 and 63 to finish in the same number of steps, 14 and 15 are the earliest duo 28, 29, and 30 is the earliest trio, and 98, 99, 100, 101, and 102 finish together as well all ending up at 22 after 10 steps.
     
  20. swiftaw macrumors 603

    swiftaw

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    Omaha, NE, USA
    #20
    I calculated the number of steps needed for each number between 1 and 10000. The results are here: http://homepage.mac.com/swiftaw/mathgame.txt


    As you can see, there are many occurrences where a group of consecutive numbers require the same number of steps.
     
  21. Doctor Q thread starter Administrator

    Doctor Q

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    #21
    The page is great, and those groupings are definitely common and quite interesting.

    Example:
    9942 73
    9943 73
    9944 73
    9945 73
    9946 73
    9947 135
    9948 73
    9949 73
    9950 73
    9951 73​
     
  22. Doctor Q thread starter Administrator

    Doctor Q

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    #22
    Ooh, here's one I like. The number 10^9 (a thousand million, or a billion in American terminology) takes 10^2 (100) steps!
    1000000000 -> 500000000 -> 250000000 -> 125000000 -> 62500000 -> 31250000 -> 15625000 -> 7812500 -> 3906250 -> 1953125 -> 5859376 -> 2929688 -> 1464844 -> 732422 -> 366211 -> 1098634 -> 549317 -> 1647952 -> 823976 -> 411988 -> 205994 -> 102997 -> 308992 -> 154496 -> 77248 -> 38624 -> 19312 -> 9656 -> 4828 -> 2414 -> 1207 -> 3622 -> 1811 -> 5434 -> 2717 -> 8152 -> 4076 -> 2038 -> 1019 -> 3058 -> 1529 -> 4588 -> 2294 -> 1147 -> 3442 -> 1721 -> 5164 -> 2582 -> 1291 -> 3874 -> 1937 -> 5812 -> 2906 -> 1453 -> 4360 -> 2180 -> 1090 -> 545 -> 1636 -> 818 -> 409 -> 1228 -> 614 -> 307 -> 922 -> 461 -> 1384 -> 692 -> 346 -> 173 -> 520 -> 260 -> 130 -> 65 -> 196 -> 98 -> 49 -> 148 -> 74 -> 37 -> 112 -> 56 -> 28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1​
     
  23. themoonisdown09 macrumors 601

    themoonisdown09

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    Georgia, USA
    #23
    25 -> 76 -> 38 -> 19 -> 58 -> 29 -> 88 -> 44 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
     
  24. Doctor Q thread starter Administrator

    Doctor Q

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    #24
    Logarithmic graph of the Age Game values for 10^9:
     

    Attached Files:

  25. SilentPanda Moderator emeritus

    SilentPanda

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    #25
    My life just got easier! I'm gonna copy off siurpeeman. :D
     

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