Hint

Think about why Sam can not be a thief.

Solution

The thief can not be Sam, because the thief told the truth, and Sam, in this case, lied. The thief can not be Joe, because, in this case, Sam told the truth, but the truth was told only by the thief. So the thief is Bill.

Hint

Think about why Sam can not be a thief.

Hint

Pay attention, if a liar is forced to repeat his answer, then it will already be true.

Solution

It is enough to ask, for example, “What would you say to me a minute ago if I asked if you have a green crocodile at home?”. When answering such a question, the knight truthfully replies the true answer, and the liar lies, retelling the false answer, i.e. he will answer with the truth.

Hint

Pay attention, if a liar is forced to repeat his answer, then it will already be true.

Hint

Try splitting the original square into unit squares.

Solution

Divide the original square into 16 $1 \times 1$ squares. Amongst the 16 squares there will always be at least one that does not contain a point – 15 points cannot be placed into 16 different squares.

Hint

Try splitting the original square into unit squares.

Solution

The two coins can only be a 5 pence coin and a 10 pence coin. If one of them is not a 5 pence coin, then it is a a ten pence coin, and the other is a 5 pence coin.

Hint

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Hint

Note that the difference between the numbers of the letters M and O does not change when adding or deleting allowed letters.

Solution

When adding or deleting allowed letters, the difference between the numbers of letters of M and O does not change. Therefore, it is equal to 1 for synonyms of the word OMM and -1 for synonyms of the word MOO. Hence, these words are not synonymous.

Hint

Note that the difference between the numbers of the letters M and O does not change when adding or deleting allowed letters.

Hint

Think about whether this person can be a knight.

Solution

Both the knight and the liar will both say about themselves: “I am a knight.” Therefore, this person is not an inhabitant of the Island.

Hint

Think about whether this person can be a knight.

Hint

Could Andrew have been first?

Solution

Let us refer to the athletes by the first letter of their names. From the second condition we have that: V, D and G finished in the order D, V, G (not necessarily right after each other). From the third condition: D, B, E finished in the order E, D, B. Since (from the first condition) between B and A there are two athletes, so the order of finishing is: E, D, B, two athletes, A. Then it is easy to determine the order: E – first, D – second, B – third, V – fourth, G – fifth, A – sixth.

Hint

Could Andrew have been first?

Solution

a$)$ Even; b$)$ the sum is even, the product is odd; c$)$ the sum is odd, the product is even.

Hint

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Hint

Pay attention, that from a bagel “to make” a sector, one cut is required.

Solution

When a bagel is cut into pieces, the number of cuts and the number of sectors coincide, since one cut is needed in order to “make” a sector from the bagel.

In 2. the number of pieces is always 1 more than the number of cuts, because the first cut divides the log into two parts, and each next cut adds another piece.

Hint

Pay attention, that from a bagel “to make” a sector, one cut is required.

Hint

Remember task 89914.

Solution

When a bagel is cut into pieces, the number of cuts and the number of sectors coincide, since one cut is needed in order to “make” a sector from the bagel.

Hint

Remember task 89914.

Hint

Into how many parts is the log divided into by the first cut? How does the number of pieces change after each subsequent cut?

Solution

The number of pieces is always 1 more than the number of cuts, because the first cut divides the log into two parts, and each next cut adds another piece.

Answer

11 pieces.

Hint

Into how many parts is the log divided into by the first cut? How does the number of pieces change after each subsequent cut?

Hint

Remember task 89912.

Solution

Jemima’s second answer differs from the first, which means that something has changed. What? Only the situation, for example, the number of questions asked or the time. Accordingly, possible questions are: “How many questions have I already asked you?”, Or: “What time is it, to the nearest second?”.

Hint

Remember task 89912.

Hint

Try to find a question related to such a characteristic that is true for one boys, and is incorrect for the other.

Solution

The question should relate to such a characteristic that is true for one boys, and is incorrect for the other. For example: “Do you always tell the truth?”.  A question can be made more complicated indefinitely, for example, “Does your friend always speak the truth?”.

Hint

Try to find a question related to such a characteristic that is true for one boys, and is incorrect for the other.

Hint

What is the interval between two consecutive chimes?

Solution

Since 3 chimes of the clock occur in 12 seconds, the interval between two consecutive strokes is 6 seconds. From the first chime to the second – 6 seconds and from the second to the third – also 6 seconds. Six chimes are distributed with 5 intervals. Consequently, 6 chimes of the clock will strike for $5 \times 6$ = 30 seconds.

Hint

What is the interval between two consecutive chimes?

Solution

999 + 999 – 9 = 1989.

Hint

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Hint

Since it is impossible to fulfill all of the instructions in the will, you must only fulfill some of them. The method you use will depend on which part of the instructions you pick.

Solution

Since it is impossible to fulfill all of the instructions in the will, we must only fulfill some of them. The method we use will depend on which part of the instructions we pick. There are many possibilities. For example:
$\\$ 1) From the first condition of the will it follows that the son must receive 2/3 of the inheritance and from the second condition it follows that the daughter must receive half the amount the mother receives.
$\\$ 2) From the first condition it follows that the mother receives half the amount the son receives and from the second condition it follows that this portion is twice the amount the daughter receives.
$\\$ 3)In each condition the mother does not receive less than a third while the son’s inheritance is 4 times that of the daughter.
$\\$ Other options are also possible. This problem came from reality. Such a situation actually happened in Ancient Rome. The inheritance was divided by a court as in case 2): 4/7 of the property went to the son (i.e. 120 talents), 2/7 to the mother (i.e. 60 talents) and 1/7 to the daughter (i.e. 30 talents).

Hint

Since it is impossible to fulfill all of the instructions in the will, you must only fulfill some of them. The method you use will depend on which part of the instructions you pick.

Solution

Redundant definition – four-nosed. In all of other definitions, the body part characteristic contains the letter “E” twice, whereas the word “nosed” has only one “E”.

Hint

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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