# Piece of cake! (Help me please)

#### absoluzation

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?

#### DarnItJimImAnEngineer

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 $\mathcal{O}(10~g/mol)$).

#### absoluzation

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?

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#### absoluzation

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

#### absoluzation

Thank you so much tho! I get it thanks to you
But wait.. my teacher told me the answer is one of the following:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9

So 10 isn't in there so I guess the unit digit isn't 10?

#### DarnItJimImAnEngineer

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.

#### DarnItJimImAnEngineer

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.)

#### Yooklid

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
The units digit (sometimes referred to as the unit digit) of 10 is 0.

#### absoluzation

The units digit (sometimes referred to as the unit digit) of 10 is 0.
So you agree with the answer being 10 but the unit digit is 0 right