108. Two Fours. I am constantly asked about the old puzzle "Four Fours." I published it in 1899, but later found out that it was first published in the first volume of the magazine Knowledge in 1881. Since then, various authors have referred to it. The puzzle is formulated as follows: "Find all possible numbers that can be obtained from four fours (no more and no less) using various arithmetic signs. For example, the number 17 can be represented as $4 \times 4 + 4 / 4$, the number 50 as $44 + 4 + \sqrt{4}$, and so on. Similarly, all numbers up to 112 inclusive can be written using only the signs of addition, subtraction, multiplication, division, square root, decimal point* and the factorial sign (for example, you can write 4!, which means just $1 \times 2 \times 3 \times 4$, or 24). The number 113 cannot already be represented as a combination of four fours.

It is necessary to determine which numbers can be written using one, two, and three fours. Major difficulties arise because some numbers are not easily represented in this way. For example, I think that very few will be able to express 64 using two fours. Will the reader be able to do this?
108. Here is how the number 64 can be expressed using two fours and arithmetic signs:

\[
\sqrt{(\sqrt{\sqrt{4}})^{4!}}=\sqrt{(\sqrt{2})^{24}}=\sqrt{2^{12}}=\sqrt{4096}=64
\]

[Interest in the "Four Fours" problem has periodically revived since its publication. I wrote about a relatively recent discussion of this problem in the January 1964 issue of Scientific American (see also the note in the answers section of the following issue of the same magazine). A table expressing all numbers from 1 to 100 using four fours can be found in the books: L. Harwood Clarke "Fun With Figures" (N. Y., 1954, pp. 51-53) and Angela Dunn "Mathematical Bafflers" (N. Y., 1964, pp. 5-8).

The number 64 can easily be expressed using four fours: \((4+4) \times(4+4)\), as well as using three fours: \(4 \times 4 \times 4\). M. Bicknell and V. E. Hoggatt in the journal Recreational Mathematics Magazine (14, 1964) list 64 ways to express 64 using four fours.

Knuth in the journal Mathematics Magazine (37, 1964, pp. 308-310) showed how to represent 64 using only one four and three types of symbols: the square root sign, the factorial sign, and parentheses. To express 64 in this way, 57 square root signs, 9 factorial signs, and 18 parentheses are required. With the help of a computer, it was determined that all positive integers up to 208 can be expressed in a similar manner. Knuth conjectures that this method is applicable to all positive integers.

Dudeney is partially correct in his statement regarding 113. As far as I know, no one has managed to represent this number without using highly non-standard symbols or extremely complex procedures, such as the one proposed by Knuth. -M. [\.