To obtain the summary of a number with up to 9 digits, you should write how many digits it has, then how many of those digits are odd, and finally how many are even. For example, the number 9103405 has 7 digits, 4 of which are odd and 3 are even, so its summary is 743.

a) Find a number whose summary is 523.

b) Find a number that is equal to its own summary.

c) For any number with up to 9 digits, we can calculate the summary of the summary of its summary. Show that this procedure always leads to the same result, regardless of the initial number.
Solution

a) Consider a number whose summary is 523. Then it has five digits (523), of which two are odd (523) and three are even (523). We can form many numbers satisfying these conditions; some examples are 11222, 23456, and 36854.

b) Since the summary of any number has three digits, we see that for a number to be equal to its own summary, it must have three digits. Suppose then that there exists a number that is its own summary, and let $c$ be its hundreds digit, $d$ the tens digit, and $u$ the units digit. Since the hundreds digit of the summary of a three-digit number is 3, we must have $c=3$. Adding the tens and units digits of the summary should give the number of digits of the original number, i.e., $d+u=3$. Therefore, the possible summaries of a summary are 303, 312, 321, or 330; of these, the only one that is its own summary is 321, which is then the number we are looking for.

c) The summary of a number always has three digits. As we saw in part $b$), the four possibilities for the summary of a three-digit number are 303, 312, 321, or 330; thus, the possibilities for the summary of the summary of any number are among these four. The summaries of 303, 312, 321, or 330 are all equal to 321. Since 321 has itself as its summary, we will always arrive at it when we calculate the summary of a number successively. More precisely, for any initial number, the summary of the summary of its summary is 321. We can visualize this reasoning in the diagram below.

number $\xrightarrow{\text{summary}}$ three-digit number $\xrightarrow{\text{summary}} 303, 312, 321$, or $330 \xrightarrow{\text{summary}} 321$

Note: We observe that the summary of the summary of any number can only be 303 or 321. Indeed, the first time we calculate the summary of some number, we get a three-digit number $c d u$, with $d+u=c$. We then have two cases:

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