If you go through the powers of 2 -- 2, 4, 8, 16, etc. -- in ascending order, the last digit encountered will be 7 (in 32768 = 2^15, the second last being the 9 in 4096). What is the last two-digit combo you'll hit and at what power of 2? (Obviously the sequence starts with 16, 32, 64, 12, 28, 25, 56,... but which one is the last to be seen?)

Hi Wes,

Well, this is not an answer...yet. I am sitting here with my 14 year old son, who can actually do math, and have turned the problem over to him. He tells me the only way to get this is just powering through it, and that there is no elegant solution to getting the answer. Apparently there is something called the equaldistribution theorem, though, that guarantees that all two digit - and in fact all any digit - numbers will eventually appear and that this is because log(10)2 is an irrational number. So I am well out over my skis here, but Jimmy says that this has something to do with the Coupon Collector's Problem - where you need to estimate the random occurrence of all coupons in boxes of cereal. According to that we should start looking for that final pair at around 2 to the power of 60. He is going to do it on the train on the way to school this morning...Dad, is going to turn his brain off again.

Oh, for reference:

Equidistribution theorem
https://en.wikipedia.org/wiki/Equidistribution_theorem

Coupon Collector's Problem
https://en.wikipedia.org/wiki/Coupon_collector%27s_problem

and right now, 2 to power of 60.
1,152,921,504,606,846,976

This is from google AI search:

The value of 2^66 is 73,786,976,294,838,206,464, and the number 78 is the final two-digit number (from 10 to 99) to appear within the sequence of the powers of 2. In recreational mathematics, this is a famous trivia fact regarding how numbers manifest within exponential expansions. If you scan the digits of the powers of 2 (2^0, 2^1, 2^2, ...) looking for two-digit substrings, you will find that 89 out of the 90 possible two-digit numbers show up early on. The number 78 is the lone holdout. When you reach 2^66 78 finally appears for the very first time (as the third and fourth digits: 737869...). This completes the entire set of two-digit numbers from 10 to 99.

The table below shows the last five two-digit numbers to make their debut in the powers of 2 sequence, along with the specific power where they first appear:

Two-Digit Power of  Full value of
Number    2 (2^n)   Power of 2
------------------------------------------
   22     2^43           8,796,093,022,208
   66     2^46          70,368,744,177,664
   50     2^52       4,503,599,627,370,496
   61     2^58     288,230,376,151,711,744
   78     2^66  73,786,976,294,838,206,464

https://tinyurl.com/78-is-last-powers-of-2
It may not show up as per quote above when you click on the link.

gtho4,

A www...I just got home and my kid got the same answer. He worked on it on the train and was happy to tell me when he got home. Too bad he got pipped by AI, but that's the future I guess. So, the win goes to the machine, but he was happy to tell me that his guess of 2 to the power of sixty was pretty close. There were only three unfound digits remaining between 56 and 66 and your AI is correct, 78 was the last one.

J

This webpage has a complete list from 2^1 through to 2^66

2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024
2^11 = 2048
2^12 = 4096
2^13 = 8192
2^14 = 16384
2^15 = 32768
2^16 = 65536
2^17 = 131072
2^18 = 262144
2^19 = 524288
2^20 = 1048576
2^21 = 2097152
2^22 = 4194304
2^23 = 8388608
2^24 = 16777216
2^25 = 33554432
2^26 = 67108864
2^27 = 134217728
2^28 = 268435456
2^29 = 536870912
2^30 = 1073741824
2^31 = 2147483648
2^32 = 4294967296
2^33 = 8589934592
2^34 = 17179869184
2^35 = 34359738368
2^36 = 68719476736
2^37 = 137438953472
2^38 = 274877906944
2^39 = 549755813888
2^40 = 1099511627776
2^41 = 2199023255552
2^42 = 4398046511104
2^43 = 8796093022208
2^44 = 17592186044416
2^45 = 35184372088832
2^46 = 70368744177664
2^47 = 140737488355328
2^48 = 281474976710656
2^49 = 562949953421312
2^50 = 1125899906842624
2^51 = 2251799813685248
2^52 = 4503599627370496
2^53 = 9007199254740992
2^54 = 18014398509481984
2^55 = 36028797018963968
2^56 = 72057594037927936
2^57 = 144115188075855872
2^58 = 288230376151711744
2^59 = 576460752303423488
2^60 = 1152921504606846976
2^61 = 2305843009213693952
2^62 = 4611686018427387904
2^63 = 9223372036854775808
2^64 = 18446744073709551616
2^65 = 36893488147419103232
2^66 = 73786976294838206464

https://benvitalenum3ers.wordpress.com/2012/01/23/powers-of-2/

My question was actually looking for two-digit combinations, which include 00 to 09, not just two-digit numbers. But using the list, it is quickly visible that the answer would be the same. All 0x combos show up before the 78.