# 7. Clone 1

In the expression OL $*$ IM $* P *$ IA * DA, it is required to replace the asterisks with two plus signs and two minus signs, and to replace the letters with digits according to the rules of a cryptarithm (identical letters with identical digits, and different letters with different digits). What is the maximum value of the expression that can be obtained in this way? The first digit in a two-digit number must be different from zero.

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# Answer: 263

## Solution

The value of the expression will be the greatest if the addends with a plus sign are as large as possible, and the addends with a minus sign are as small as possible. Pluses should be placed before the two-digit numbers IM and IP, then we will get the maximum possible sum because we can use the maximum digit 9 twice in the tens place.

Next, we maximize the value of the expression

$$
\text{OL} + \text{IM} - P + \text{IA} - DA = 10 \cdot (2I + O - D) + (M + L - P).
$$

From the last equality, it is clear that I is multiplied by 20, O by 10, and M and L by 1. To get the maximum sum, the largest digit should be multiplied by the largest number. Then \( I = 9, O = 8, M = 7 \) and \( L = 6 \) (we can take 6 and 7 in the opposite order). Similarly, we get the digits with a minus sign. The smallest should be D, as it is multiplied by 10. But D cannot be zero, as it is the first digit of a two-digit number. Then \( D = 1, P = 0 \). The digit A can be any of the remaining digits, as it does not affect the result. Let, for example, \( A = 5 \). The maximum value of the expression will be \( 86 + 97 - 0 + 95 - 15 = 263 \).

## Clone 2

In the expression \(\mathrm{KP} * \mathrm{OC} * \mathrm{CB} * \mathrm{OP} *\) D, two plus signs and two minus signs need to be placed instead of the asterisks, and the letters need to be replaced by digits according to the rules of a cryptarithm (identical letters - identical digits, different letters - different digits). What is the greatest value of the expression that can be obtained in this way? The first digit in a two-digit number must be different from zero.

Answer: 263

## Clone 3

In the expression \(\mathrm{K} * \mathrm{AR} * \mathrm{TI} * \mathrm{NN} * \mathrm{AY}\), two plus signs and two minus signs need to be placed instead of the asterisks, and the letters need to be replaced by digits according to the rules of a cryptarithm (identical letters - identical digits, different letters - different digits). What is the greatest value of the expression that can be obtained in this way? The first digit in a two-digit number must be different from zero.

Answer: 170