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The Age Game (math challenge)

Doctor Q

Doctor Q

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Original poster
Staff member
Sep 19, 2002
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Los Angeles
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?​
 
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
 
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.
 
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
 
Doctor Q said:
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.​

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?​
Click to expand...

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

Doctor Q said:
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?
Click to expand...
Shall I start with 3E+25 and let you know? :p
 
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.
 
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​
 
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
 
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.
 
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
 
atszyman said:
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.
Click to expand...

Curses!
I was trying to find an excel formula for that!
 
I think I did this correctly.

46-23-70-35-106-53-160-80-40-20-10-5-16--8-4-2-1
 
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.
 
tkidBOSTON said:
Curses!
I was trying to find an excel formula for that!
Click to expand...

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.
 
swiftaw said:
As you can see, there are many occurrences where a group of consecutive numbers require the same number of steps.
Click to expand...
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​
 
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​
 
Logarithmic graph of the Age Game values for 10^9:
 

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