Let's find an odd natural number whose first six powers, when written out, use 61 digits, each of the six powers starts with a different digit, and the sum of the leading digits is 24![^0]


[^0]:    ${ }^{1}$ It can help if we plot the power functions on graph paper with logarithmic scales on the axes, meaning we measure segments on the axes proportional to the logarithm of the base, or the exponent.
I. Using the data about the 61st digit, we obtain lower and upper bounds for the base-10 logarithm of the sought number $N$. Let the characteristic of $\lg N$ be $k$ (clearly $k \geqq 0$), and its mantissa be $m$. According to this,

$$
10^{k} \leqq N < 600$, and by further increasing $N$, only a single power's leading digit can increase by 1 to compensate for the still existing 24-23 deficit.

![](https://cdn.mathpix.com/cropped/2024_05_02_db2b3260f0df7e731bd2g-2.jpg?height=670&width=642&top_left_y=126&top_left_x=725)

Considering the second digit of these powers, it is expected that the leading digit of $N^{5}$ and $N^{6}$ will reach 8 and then 9, and 5, respectively. (We also considered that the growth is faster for larger exponents.) Therefore, we calculate - taking (2) into account - the fifth root of the transition points for $8 \cdot 10^{13}$ and $9 \cdot 10^{13}$, and the sixth root of $5 \cdot 10^{16}$. These (from their logarithms, which are respectively $2.7806$, $2.7908$, and $2.7832$):

$$
603.4, \quad 617.7, \quad 606.9,
$$

so the sum of the leading digits jumps to 24 at 604 and to 25 at 607. Among these, the only odd number is $N=605$. Indeed, in standard form,

$$
\begin{array}{ll}
605^{2}=3.66 \cdot 10^{5}, & 605^{3}=2.21 \cdot 10^{8}, \quad 605^{4}=1.34 \cdot 10^{11}, \\
605^{5}=8.10 \cdot 10^{13}, & 605^{6}=4.90 \cdot 10^{16},
\end{array}
$$

and here the leading digits are different, so the sought number is indeed 605. (We used the requirement of different leading digits only for verification.)

Based on the works of Galántai Aurél (Budapest, Bem J. Industrial Secondary School, 4th grade), Máté András (Budapest, I. István Gymnasium, 2nd grade), and Balogi Zoltán (Debrecen, Fazekas M. Gymnasium, 1st grade), with additional comments.

![](https://cdn.mathpix.com/cropped/2024_05_02_db2b3260f0df7e731bd2g-2.jpg?height=886&width=1092&top_left_y=1617&top_left_x=495)

Note. The suggested plot is simple because the image of the function $y=x^{n}$ on the recommended, so-called double logarithmic function paper is a straight line, since $\lg y=n \cdot \lg x$. The leading digit of $x^{n}$ can be read from the 1st figure for $1 \leqq x \leqq 10$ and $n=1,2,3$. The 2nd figure provides these for the (3) interval on a different scale, for all 6 exponents, thus we can perform the verification in the solution through a different method.