Question 3

If $$X$$ is a 1000 digit number, $$Y$$ is the sum of its digits, $$Z$$ the sum of the digits of $$Y$$ and $$W$$ the sum of the digits of $$Z$$, then the maximum possible value of $$W$$ is

Solution

Since $$X$$ has 1000 digits, $$Y \leq 9000$$, so $$Z$$ can be any digit-sum achievable by a number up to 9000, with the highest such digit-sum being 29 (for example when $$Y=2999$$). Taking $$Z=29$$ gives $$W=2+9=11$$, which is the maximum since no value up to 35 (the true maximum of $$Z$$) has a higher digit sum than 29.

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If $$X$$ is a 1000 digit number, $$Y$$ is the sum of its digits, $$Z$$ the sum of the digits of $$Y$$ and $$W$$ the sum of the digits of $$Z$$, then the maximum possible value of $$W$$ is

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