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First off, this is a badly titled question because I'm unsure of how to word the problem. Please suggest a better title.

The sum of $55,555$ and $33,333$ is $88,888$. If I change the first digit of each number, the maximum that their sum can change is from the 1s to the 10s digits.

$$ 55,555 + 33,333 = 88,888 $$ $$ 55,559 + 33,339 = 88,898 $$

We can see that the first three digits of the sum remain the same. The first two, the 10s digit changes. The 1s digit can also change, but it did not in this case.

This pattern repeats again if we change the first two digits. To summarise, I made this graphic.

enter image description here

Is there a mathematical term or equation to describe this pattern?

Mohamad
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1 Answers1

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We can generalize this pattern as follows. Given the lenght $n$ and digits $a$, $b$, $a’$, and $b’$ in the sum $$S_0=(10^na+\dots+10a+a)+$$ $$(10^nb+\dots+10b+b)$$ (your example corresponds to $n=5$, $a=3$, $b=5$, $a’=9$, and $b’=9$) we concecutively replace $a$ by $a’$ and $b$ by $b’$. That is for $1\le i\le n$

$$S_i=(10^na+\dots+10^{i}a+10^{i-1}a’+\dots 10a’+a’)+ (10^nb+\dots+10^{i}b+10^{i-1}b’+\dots 10b’+b’).$$

Then $$S_i-S_{i-1}=10^{i-1}a’-10^{i-1}a+10^{i-1}b’-10^{i-1}b=10^{i-1}(a’-a+b’-b).$$

In your example $$10^{i-1}(a’-a+b’-b)=10^{i}.$$

Alex Ravsky
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