## February 2022 - Challenge

It is well known that every natural number can be uniquely written as the sum of powers of 2, e.g., 13 = 1 + 4 + 8. The same goes for powers of 3, e.g., 13 = 1 + 3 + 9. In these representations, for every pair of summands, one divides the other, i.e., 4 divides 8 and 3 divides 9.

However, if we allow the representation of numbers as a sum of the **product** of a power of 2 and a power of 3, it is possible to find for every number a representation where for each pair of summands, none divides the other, e.g., 13 = 4 + 9, 15 = 2\cdot3 = 6 + 9, and as a more complex example, 719 = 32 + 48 + 72 + 324 + 243. In the last example, the list of powers involved can be also written as [(5, 0), (4, 1), (3, 2), (2, 4), (0, 5)], where each pair (a,b) represents the number 2^{a}3^{b}.

Let’s look at the number 10100100101110110000. That’s a 20-digit number in base 10, whose digits consist only of 1s and 0s, with a leading 1. It can be represented in the above method as [(41, 2), (30, 12), (29, 13), (26, 19), (20, 23), (13, 28), (12, 31), (4, 37)].

**Your goal**: Find a number with exactly 200 decimal digits, consisting only of the digits 1 and 0 with a leading 1, that can be represented by a sum of less than 150 summands. Write your solution in two lines: The first consisting of the number itself, and the second consisting of the list of powers, in the list format given above.

**A bonus** "*" will be given for finding a 300-digit number with a sum of less than 215 summands.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

*If you have any problems you think we might enjoy, please send them in. All replies should be sent to:* ponder@il.ibm.com

**Challenge:**
01/02/2022 @ 12:00 PM EST

**Solution:**
04/03/2022 @ 12:00 PM EST

**List Updated:**
07/02/2022 @ 12:00 PM EST

#### People who answered correctly:

***Lazar Ilic **(1/2/2022 2:29 PM IDT)

***Paul Revenant **(1/2/2022 4:51 PM IDT)

***Gaétan Berthe **(1/2/2022 5:05 PM IDT)

**Reiner Martin **(2/2/2022 1:48 AM IDT)

***Sanandan Swaminathan **(2/2/2022 8:47 AM IDT)

***Anders Kaseorg **(2/2/2022 2:16 PM IDT)

***Alexander Gramolin **(3/2/2022 3:23 AM IDT)

***Benjamin Lui **(3/2/2022 8:16 AM IDT)

***Uoti Urpala **(3/2/2022 9:17 AM IDT)

***Vladimir Volevich **(3/2/2022 7:57 PM IDT)

***Bert Dobbelaere **(5/2/2022 11:36 AM IDT)

***Asaf Zimmerman **(6/2/2022 5:48 PM IDT)

***Latchezar Christov **(7/2/2022 10:50 PM IDT)

***Peter Moser **(8/2/2022 4:00 PM IDT)

***Amos Guler **(8/2/2022 11:34 PM IDT)

***Oleg Vlasii **(9/2/2022 10:27 AM IDT)

***Daniel Chong Jyh Tar **(9/2/2022 5:48 PM IDT)

***Marco Bellocchi **(11/2/2022 7:56 AM IDT)

***Li Li **(11/2/2022 11:29 AM IDT)

***Andreas Stiller **(11/2/2022 8:09 PM IDT)

***Pål Hermunn Johansen **(12/2/2022 5:53 PM IDT)

***Daniel Bitin **(12/2/2022 11:15 PM IDT)

***Gary M. Gerken **(13/2/2022 6:51 AM IDT)

**Maksym Voznyy **(13/2/2022 8:31 PM IDT)

***Thomas Egense **(15/2/2022 4:45 PM IDT)

***Alper Halbutogullari **(15/2/2022 9:51 PM IDT)

***Bertram Felgenhauer **(17/2/2022 3:47 PM IDT)

***Dan Dima **(17/2/2022 7:52 PM IDT)

***Franciraldo Cavalcante **(19/2/2022 12:27 AM IDT)

***Tim Walters **(21/2/2022 3:20 AM IDT)

***David Greer **(21/2/2022 1:28 PM IDT)

***Liubing Yu **(23/2/2022 9:52 AM IDT)

**Peter Wu **(23/2/2022 11:32 AM IDT)

***Chris Shannon **(24/2/2022 2:58 AM IDT)

***Karl Mahlburg **(24/2/2022 4:41 AM IDT)

**Eran Vered **(25/2/2022 6:24 PM IDT)

***Stephen H. Landrum **(27/2/2022 12:26 AM IDT)

***Motty Porat **(27/2/2022 1:37 AM IDT)

***John Goh **(1/3/2022 5:42 AM IDT)

***Radu-Alexandru Todor **(1/3/2022 10:58 AM IDT)

**Nyles Heise **(3/3/2022 5:17 AM IDT)

***Clive Tong **(3/3/2022 7:15 PM IDT)