Solution

From the first equation, it follows that $M = 1,$ and from the second, that $A$ does not exceed $3,$ then $A = 2$ or $A = 3.$ Since $12 \times 12 = 144,$ then we get from the first equation that $I = R,$ which is a contradiction. The case $A = 3$ gives the answer: $13 \times 13 = 169$ and $31 \times 31 = 961.$

Hint

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Solution

When multiplying the divisor by 8, we get a two-digit number. This means that the divisor does not exceed 12, because 13 × 8 = 104 is a three-digit number. If the divisor is multiplied by the first or last digit of the quotient $($which is greater than 8$)$, we get a three-digit number, hence the divisor is 12. So, 12 × 989 = 11868.

Answer

11868/12 = 989.

Hint

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Solution

Because $4 \times M$ is even, then H is an even number. Then $H = 2,$ because otherwise the product on the left would give a six-digit number. Therefore, $A = 8$ or $A = 9,$ and $M = 3$ or $M = 8.$ Since the product on the left gives a five-digit number, then $A ≤ 4.$ Let $M = 3$ since $4 \times H + 1$ ends in $A$ then $A$ is odd that is $A = 1$ then $H = 5,$ but then $A = 8$ or $A = 9$ do not fit. Let $M = 8$ and then $A = 8;$ the number $4 \times I + 3$ ends in a $A$ then $A$ is an odd number, i.e. $A = 1$ or $A = 3.$ If $A = 1$ then $A$ is not equal to $9.$ Consequently $A = 3$ and $H = 5.$

Answer

$23958 \times 4 = 95832.$

Hint

Solution

Ten letters are used and, consequently, all numbers. It is clear that the largest of the digits is T, hence T = 9. Removing from the inequalities T, we see that the largest of the remaining digits is I. And so on.

Answer

9 $>$ 7 $>$ 6 $>$ 0 $<$ 1 $<$ 2 $<$ 3 $<$ 7 $<$ 9 $>$ 8 $>$ 7 $>$ 3 $<$ 4 $<$ 5 $<$ 6

Hint

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Hint

From the form of the first digits of all three numbers it follows that $K < I.$ Why?

Solution

For convenience of presentation, we write this puzzle as follows.

From the first column, it is clear that $K < I.$ From this and from the last column it follows that $S + I = K + 10$ $($and not $S + I = K)$. Then from the second column we deduce $1 + I + S = S,$ or $1 + I + S = 10 + S.$ The first option is impossible, and from the second we immediately determine that $I = 9.$ Hence $K = 4, S = 5.$ The entire puzzle is decoded as follows: $495 + 459 = 954.$

Answer

$495 + 459 = 954.$

Hint

From the form of the first digits of all three numbers it follows that $K < I.$ Why?

Solution

From the puzzle, it is clear that $A + C = 10,$ $A + B + 1 = C + 10,$ $A + 1 = B.$ Solving this system, we get: $A = 6, B = 7, C = 4.$

Answer

$6 + 67 + 674 = 747.$

Hint

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Hint

What is C?

Solution

Since there is one two-digit and two single-digit numbers among the summands in the puzzle, and the sum is a three-digit number, this three-digit number must begin with 1, and the two-digit number starts at no less than $8.$ So $C = 1, B = 8$ or $9.$ If $B$ was $8,$ then $C$ for any $A$ would have to be even $($see the last column$)$.

But $C = 1,$ then $B = 9.$ Knowing $B$ and $C$ we determine that $A = 6.$ So, when replacing letters with numbers, we get: $6 + 99 + 6 = 111.$

Answer

A = 6; B = 9; C = 1.

Hint

What is C?

Solution

In the first table, on each line in the first column we have the basis of the degree, in the third column we have the exponent and in the second the result of the exponentiation. Thus, the missing number is 49.
The second table is constructed differently: there are pairs of equal numbers but one number is written in the form of a decimal fraction, and the other in the form of an ordinary fraction. Thus, the extra number is 5/13.

Answer

49; 5/13.

Hint

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Hint

Think about what the first digit of the phone number could mean? Which two numbers are obtained from the remaining three digits?

Solution

The first digit of the phone number is equal to the number of letters in the surname, and the last three are to the numbers in the alphabet of the first and last letters of the surname. It follows that Ognef’s telephone number is 5156.

Answer

5156

Hint

Think about what the first digit of the phone number could mean? Which two numbers are obtained from the remaining three digits?

Solution

If the number 2 in the number 102 is moved upwards, to the place of the exponent, then the original equality takes the form 101 – $10^2$ = 1 and will be true.

Answer

The number 2 in the number 102 should be put in the place of the exponent.

Hint

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Hint

In the first 2 circles, consider which pairs of letters words can start with.

Solution

In the first circle, the arrow must be put on the letter B, for words cannot start with the combinations $JK$, $JQ$, $XK$, $XQ$, $JL$ and $XL.$

In the second circle, the arrow must be placed on the letter L, since words can only start with $BL$ and not with $BK$ or $BQ$.

Thus, the first two letters of the word are $BL.$
The third circle is the only one with a vowel in it so the arrow in this circle must be put on the letter O.
For the fourth circle, we consider the possible words that can be formed. These are: $BLOF,$ $BLOH$ and $BLOT.$ The only real word out of these three is $BLOT.$

Answer

$BLOT$

Hint

In the first 2 circles, consider which pairs of letters words can start with.

Hint

Do the key elements not remind you of small fragments of the main picture?

Solution

The key shows which arrows depart from the location of a letter which we must choose. As a result, the we get the word COMPUTERS.

Answer

COMPUTERS

Hint

Do the key elements not remind you of small fragments of the main picture?

Hint

What letter is zero encrypted as? Can the letter “a” denote an odd number $($see the last line in the figure for the task$)$? Can “a” be equal to 2 $($see the first line in the figure$)$?

Solution

Consider the drawing.

Hint

What letter is zero encrypted as? Can the letter “a” denote an odd number $($see the last line in the figure for the task$)$? Can “a” be equal to 2 $($see the first line in the figure$)$?

Hint

Can both sides of the equation be multiples of 7?

Solution

The equality cannot be true because one of the letters must necessarily correspond to the number 7; then that part of the equality into which this letter enters will be divisible by 7, and the second part of the equality will not be divisible by 7. Hence, they cannot be equal. This reasoning is also valid for the number 5.

Answer

This is impossible: one part of the equation will be a multiple of 7, the other not.

Hint

Can both sides of the equation be multiples of 7?

Hint

Note: the first letter is either B or V.

Solution

To solve this problem, we write down the letters with the numbers 1, 2, 5, 6, 12, 21, 22, 26. These will be, respectively, “A”, “B”, “E”, “F”, “L”, “U”, “V”, “Z”. Now let’s look at the options. The first letter is either “B” or “V”. In the first case, further variants of “BBBA”, “BBU”, “BVA”, “BBU”, “BBUL”, “BVL” are possible. No English word begins with such letters. Hence, the first letter in the word will be “V”. Let’s continue. Further, the letter combinations can now be “VBUB”, “VBAB”, “VBAV”, “VUBB”, “VUVU”, “VUVA”. Among these combinations, only “VUVU” gives hope for finding the desired word. Continuing the search, we get the only possible answer: “VUVUZELA”.

Answer

VUVUZELA

Hint

Note: the first letter is either B or V.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $(in our case, from 2 to 9)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1, k, k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.

Solution

When solving these problems, some general considerations can be used. For example, the number 2 can be represented in the form $(m/m)$ + $(m/m)^m$, where m is any number $($in our case, from 2 to 9$)$. We also note that if the number k can be represented using only 3 digits m, then the numbers $k – 1,$ $k,$ $k + 1$ can be represented by subtracting, multiplying or adding the obtained three-digit number and $m/m.$

Hint

Try to write a formula which, when you insert any five identical digits into it, you get 1.