Piece of cake! (Help me please)

Dec 2019
52
1
ok
Ruprecht has prepared a huge cake and invited 100 elves to a cake eating party. The elves are numbered from 1 to 100 and visit Ruprecht one by one in the afternoon.

  • The elf with number 1 receives 1 % of the cake.
  • The elf with number 2 receives 2 % of the rest of the cake.
  • The elf with number 3 receives 3 % of the rest of the cake.
  • The elf with number 4 receives 4 % of the rest of the cake.
  • And so on.
  • The elf with number k receives k % of the rest of the cake.
  • And so on.
  • The elf with number 99 receives 99 % of the rest of the cake.
  • The elf with number 100, finally, receives 100 % of the rest of the cake and hence all the remaining cake.
The elf with number N has received the largest piece of cake. What is the unit digit of N?

Please give me the calculations
 
Jun 2019
493
262
USA
Let $f_k$ be the fraction of the original cake that elf $k$ receives.
$\displaystyle f_1 = \frac{1}{100}$
$\displaystyle f_k = \frac{k}{100}\left(1-\sum_{i=1}^{k-1}f_i\right), k\in\{2, 3, …, 100\}$

$\rightarrow f_2 = \frac{2}{100}\frac{99}{100} = \frac{198}{10^4}$
$\rightarrow f_3 = \frac{3}{100}\frac{9802}{10^4} = \frac{29406}{10^6}$

Maximum occurs at $f_{10} \approx 0.0628 =6.28 \%$.

Incidentally, the pieces decrease by more than a factor of 10 each elf towards the end. The original cake would have to be approximately as massive as the Earth for elf 100 to receive a single atom (assuming atomic mass $\mathcal{O}(10~g/mol)$).
 
Dec 2019
52
1
ok
Let $f_k$ be the fraction of the original cake that elf $k$ receives.
$\displaystyle f_1 = \frac{1}{100}$
$\displaystyle f_k = \frac{k}{100}\left(1-\sum_{i=1}^{k-1}f_i\right), k\in\{2, 3, …, 100\}$

$\rightarrow f_2 = \frac{2}{100}\frac{99}{100} = \frac{198}{10^4}$
$\rightarrow f_3 = \frac{3}{100}\frac{9802}{10^4} = \frac{29406}{10^6}$

Maximum occurs at $f_{10} \approx 0.0628 =6.28 \%$.

Incidentally, the pieces decrease by more than a factor of 10 each elf towards the end. The original cake would have to be approximately as massive as the Earth for elf 100 to receive a single atom (assuming atomic mass $\mathcal{O}(10~g/mol)$).
So basically what you're saying is that the unit digit of N is 10?
 
Last edited:
Dec 2019
52
1
ok
Let be the fraction of the original cake that elf receives.


Maximum occurs at .

Incidentally, the pieces decrease by more than a factor of 10 each elf towards the end. The original cake would have to be approximately as massive as the Earth for elf 100 to receive a single atom (assuming atomic mass ).
Thank you so much tho! I get it thanks to you :)
 
Jun 2019
493
262
USA
I'm saying $N=10$. I'm not sure what the "unit digit," would mean, except maybe the digit in the ones place, which is zero.
 
Jun 2019
493
262
USA
Incidentally, I made a mistake in my $f_3$ value. One atom of cake to whomever finds it first. (I believe my $f_{10}$ value is correct, however.)
 
Jul 2008
5,248
58
Western Canada
I agree with DarnItJim, but I took the lazy approach and used a spreadsheet, getting a value of 6.28156509555295 for elf 10. If the answer is supposed to be in the range 1 to 9, then either your teacher is wrong, or else the problem has not been stated correctly.
 

skipjack

Forum Staff
Dec 2006
21,479
2,470
The units digit (sometimes referred to as the unit digit) of 10 is 0.