1. When 4444⁴444 is written in decimal notation, the sum of its digits is A. Let B be the sum of the digits of A. Find the sum of the digits of B. (A, B are written in decimal notation.)

First, let me say this: whenever you see "sum of digits in decimal notation", think: mod 9. That's because if A=k (mod 9) then the sum of digits of A will also be equivalent to k (mod 9). That's because whenever you add 1 to A, you increase one of its digits by 1, and if there's a carry, you reduce one or more of its digits by 9. The net change to the sum of the digits is always +1 (mod 9).

Now, let Z = 4444⁴444,

and let A = SumOfDigits(Z),

and let B = SumOfDigits(A),

and let C = SumOfDigits(B).

The puzzle asks us to find C.

Let's work in mod 9. 4444 = 7 (mod 9), and 7³ = 343 = 1 (mod 9), so 4444³ = 1 (mod 9)

Z = 4444⁴⁴⁴⁴ = 4444^(3*1481+1) = (4444³)¹⁴⁸¹*4444
= 1¹⁴⁸¹*7 = 7 (mod 9)

Suppose C is 13 or larger. Then, since 4+9=13, B is at least 49. Then, since 4+9+9+9+9+9=49, A is at least 499999. Then, since
4+9+9+...+9=499999 (there are 55,555 9's in that sum), A is at least 10⁵5555, which is 100000¹1111, which is larger than 4444⁴444.

So C must be smaller than 13.

Remember Z = 7 (mod 9), and Z = A = B = C (mod 9), so C = 7 (mod 9).
And since C is smaller than 13 (and larger than 0), it follows that C must be 7.

2. What is the largest number that can be obtained as the product of positive integers that add up to 100?

3³²*2² = 7,412,080,755,407,364. First, it is clear that the numbers should be as similar as possible.
That's because, given x+y is constant, the product xy is maximized when x=y.
If the numbers weren't restricted to just integers, then the problem would be to maximize f(x)=x^100/x, so I would start by finding its derivative.

f'(x)=x^100/x(100/x²)(1-ln(x))

Letting f'(x)=0, and solving for x, I see that log(x)=1, so x=e. But we're restricted to integers, and the closest integers to e are 2 and 3. I see that 3² > 2³, so I will try to use as many 3's as possible. 3³³*1 is the answer that comes to mind first, but I see that the 3*1 at the end of that can be replaced by 2*2 to increase the value, so my final answer is 3³²*2².

More about this puzzle:

More about question 2. e^100/e is about 9,479,842,689,868,760, which is only about 28% bigger than the largest value obtained by multiplying integers together. Note that 2.7*37=999, so a relatively close approximation to e^100/e is 2.7³⁷, which is 3¹¹¹/10³⁷, or 9,129,758,166,511,361.1259115979754590511595360241199911147

More about question 1. Is 7 really the right answer to the first problem? Using a high-powered calculator, I found that 4444⁴444 is:

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93242691427557719645257656471723055812922503746623

90106143667681610429029041195604156544722453982341

53378437189072083149763584631301136544692903363054

97409416174328035261292190893517872125591526291255

13240223959733696389407503366347400147766694426415

39386607106732040986806691541249305380199302700473

06104736552258986276965824647429655406555391473259

06404433102622223679309020807145492720454009830977

08233622342518619774212788950890705639865993324699

31277104734673626799143422660333436774839058609616

83867911949216631258560268654629954816391841992863

67450171422599541873555868876360035680842412214786

21695307152384094531375297756356083583426234545493

92629561201761852794970846395294550517324778732542

29944676787432793704168268693472459216355092447410

66157981696

The sum of these 16211 digits is 72601 = A. The sum of 7+2+6+0+1 is 16 = B, so C is 7.