A stoneboard was found on the territory of the ancient Greek Academia as a result of archaeological excavations.
The archeologists decided that this stoneboard belonged to a mathematician who lived in the 7th century BC. The list of unsolved problems was written on the stoneboard. The archaeologists became thrilled to solve the problems but got stuck on the fifth. They were looking for a 10-digit number. The number should consist of only different digits. Moreover, if you cross any 6 digits, the remaining number should be composite. Can you help the archeologists to figure out the answer?
Divide 15 walnuts into four groups, each group consisting of a different whole number of nuts.
Looking back at Example 12.1 what if we additionally require all differences to be less than the smallest of the three numbers?
(a) Divide 55 walnuts into four groups consisting of different number of nuts.
(b) Divide 999 walnuts into four groups consisting of different number of nuts.
George knows a representation of number “8” as the sum of its divisors in which only divisor “1” appears twice: \[8=4+2+1+1.\] His brother showed George that such representation exists for number “16” as well: \[16=8+4+2+1+1.\] He apologies for forgetting an example considering number “32” but he is sure once he saw such representation for this number.
(a) Help George to work out a suitable representation for number “32”;
(b) Can you think of a number which has such representation consisting of 7 terms?
(c) Of 11 terms?
(d) Can you find a number which can be represented as a sum of its divisors which are all different (pay attention that we don’t allow repeating digit “1” twice!)?
(e) What if we require this representation to consist of 11 terms?
George claims that he knows two numbers such that their quotient is equal to their product. Can we believe him? Prove him wrong or provide a suitable example.
In the context of Example 14.2 what is the answer if we have five numbers instead of four? (i.e., can we get four distinct prime numbers then?)
Now George is sure he found two numbers with the quotient equal to their sum. And on top of that their product is still equal to the same value. Can it be true?
A maths teacher draws a number of circles on a piece of paper. When she shows this piece of paper to the young mathematician, he claims he can see only five circles. The maths teacher agrees. But when she shows the same piece of paper to another young mathematician, he says that there are exactly eight circles. The teacher confirms that this answer is also correct. How is that possible and how many circles did she originally draw on that piece of paper?
A group of three smugglers is offered to smuggle a chest full of treasures across the dangerous river. The boat they possess is old and frail. It can carry three smugglers without the chest, or it can carry the chest and only two smugglers. The price for this job is extremely high, and the gang is more than interested in completing the job. Think of a strategy the smugglers should follow to successfully transit the chest and themselves to the other shore.