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Questions tagged [algebra]

A mathematical puzzle centered around the manipulation and solution of equations.

2
votes
2answers
360 views

Can the sum, difference and product of 2 numbers be perfect squares? [closed]

If we take 2 numbers $x$ and $y$ such that $x>y>0$ and , can $x + y$, $x - y$ and $xy$ all be perfect squares?
3
votes
2answers
78 views

Can you have the harmonic, geometric, arithmetic and quadratic mean of 2 numbers all being integers? [closed]

I created this problem looking at the hm-gm-am-qm inequality (hm = $\frac{2xy}{x + y}$, gm = $\sqrt{xy}$, am = $\frac{x + y}{2}$, qm = $\sqrt{\frac{x^2 + y^2}{2}}$ and hm $\le$ gm $\le$ am $\le$ qm). ...
13
votes
1answer
312 views

A small team doing their job

Find the unique solution to $$ 5\,ninja + retook + quarter + turn = \_\,\_\,\_\,\_ + \_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_\,\_ $$ where each unknown is a different _ _ _&#...
8
votes
2answers
368 views

Houses and Telephone #s

There are 16 houses consecutively numbered 1-16. All houses have 7-digit telephone numbers, numbered consecutively, in the same order. For example, if house #3's phone number is 1111111, house #4 will ...
4
votes
1answer
380 views

Analyzing Single-Elimination Tournaments

I know there are many questions/answers about single elimination tournaments out there, but I wanted to approach this question in a more holistic way. For a single-elimination tournament with N ...
4
votes
2answers
261 views

Trickier hand tiling

This should be a little trickier. Using $20$ rectangles of aspect ratio $1:2$, make two squares. Use sizes 2,3,7,12,16 in both squares. And split this list among ...
0
votes
1answer
178 views

Make 100€ out of 1€

With the following trick you can turn 1€ into 100€: $$ \begin{align} 1 \, € &= 2 \, € \cdot 0.5 \, € \\ &= 200 \, \text{ct} \cdot 50 \, \text{ct} \\ &= 10 \, 000 \, \text{ct} = 100 \, € \...
7
votes
2answers
943 views

Videogame salesmen

Two gamers sell their collection of XBOX games. They've got as many Euros as there were games in the collection originally for each game. They split the money by having the first gamer take 10 Euros ...
3
votes
2answers
719 views

Inductive or Deductive Reasoning [closed]

Russell wanted to impress his coworkers. At a meeting, he announces that he can predict anyone's birthday if they tell him their birthday number. To get this famous birthday number, he asked everyone ...
-1
votes
1answer
99 views

Finding the smallest number [closed]

the sum of two positive numbers equals the difference of the squares of the two numbers, which equals the quotient of the larger number when divided by the smaller. What is the smaller number? Provide ...
4
votes
3answers
540 views

What numbers fill the blanks in 3*3 grid? [closed]

I have a puzzle that I did not know how it could be solved Which numbers should be filled on the blank spaces ? I only figure out that 4 + 20 = 24 and 1 + 20 = 21 then 8 + 20 = 28 I am not sure ...
1
vote
2answers
375 views

Calculating column number based on Excel alpha column reference [closed]

Here's a super fun brain teaser!!! Can you give me a mathematical equation to calculate the column number of a Microsoft Excel column, given the alphabetic column reference? Let X, Y, & Z ...
3
votes
3answers
422 views

Determine the cost of the book [closed]

Two sisters, Saraha and Jane, go to a book store to buy an algebra book. Saraha is 10 dollars short for the book, and Jane is 1 dollar short. However, if the sisters combine their money, they still ...
7
votes
1answer
419 views

Where does the math club meet?

A puzzle from John Conway and Pizza Hut. (The competition has ended, so I think it's okay to post the puzzle here.) Our school’s puzzle-club meets in one of the schoolrooms every Friday after ...
5
votes
2answers
248 views

Iterative Floors and Ceilings

$\left\lfloor\dfrac{2016}{\left\lfloor\dfrac{2016}{k}\right\rfloor}\right\rfloor=k\quad;\qquad \left\lceil\dfrac{2016}{\left\lceil\dfrac{2016}{k}\right\rceil}\right\rceil=k$ An integer $k$ with $1\le ...
5
votes
1answer
168 views

What is the ratio between small and big x-gons?

I have been checking Gamow's question and noticed something! If we draw all diagonals there is a little polygon in the middle of the original polygon if it is odd-gon (odd-gon is defined as the number ...
11
votes
1answer
378 views

The adventitious 18-gon

Today I have drawn a regular $18$-gon on a piece of paper. My drawing shows the $18$ vertices of the polygon labeled as $P_1,P_2,\ldots,P_{18}$ in clockwise order, and it also shows all $135$ ...
11
votes
3answers
326 views

Blackboard problem with polynomial

At the beginning the blackboard contains $n$ real numbers, one of which is $0$. In every step, we may take any polynomial such that all its coefficients are currently on the blackboard, we compute all ...
10
votes
2answers
736 views

An ugly formula [closed]

Our teacher wrote the following ugly algebraic formula on the blackboard $$(n+\sqrt{n^2-1})^{4/3}+(n+\sqrt{n^2-1})^{-4/3}$$ Our teacher told us that for some positive integers $n$ this ugly formula ...
14
votes
5answers
2k views

A man bought several liters of wine

Diophantus of Alexandria was a Greek mathematician who lived in the third century (AD). Diophantus wrote a series of books called "Arithmetica", most of which are lost. Among many other problems, the ...
19
votes
3answers
1k views

Which is larger? $\sqrt{15} - \sqrt{7} + \sqrt{5} + \sqrt{2}$ versus $5$

Which side is larger? $$ \sqrt{15} - \sqrt{7} + \sqrt{5} + \sqrt{2} \stackrel{?}{\lessgtr} 5 $$ Without using a calculator, computer, or estimating square roots, please determine which side has the ...
7
votes
3answers
471 views

Six fruit baskets

There are six fruit baskets with apples, bananas and plums on the table. The number of bananas in each basket is equal to the total number of apples in the other five baskets together. The number of ...
8
votes
2answers
244 views

Cheherazad, the carpet salesman

Cheherazad, the carpet salesman, has bought a rectangular piece of carpet. Unfortunately, Cheherazad's has lost his tape measure and he has no other measuring instruments. However, he finds that if he ...
12
votes
5answers
3k views

Turning wine into water

Niccolò Tartaglia was an Italian mathematician, engineer and bookkeeper, living in the 16th century in the Republic of Venice. Tartaglia published many books, including his "General Trattato di Numeri"...
12
votes
2answers
2k views

The grazing cows of Sir Isaac Newton

Sir Isaac Newton's book "Arithmetica Universalis" contains the following famous puzzle: In $4$ weeks, $12$ cows graze bare $3\frac13$ acres of pasture land, and in $9$ weeks, $21$ cows graze ...
5
votes
4answers
412 views

And the average is 51.5625

Freddy removes four consecutive even numbers from the set $\{1,2,3,\ldots,n\}$. Freddy then computes $51.5625$ as the average of the remaining $n-4$ numbers. Which four even numbers did Freddy remove?...
21
votes
2answers
864 views

The year 2016 is approaching

Do there exist integers $r,s,t\ge100$ such that the decimal representation of $(r+\sqrt{s})^t$ is of the form $~~~\ldots\ldots2015\,.\,2016\ldots\ldots$? (In other words: the four digits before the ...
2
votes
4answers
628 views

Fifteen white rhinoceroses

Fifteen white rhinoceroses are standing in a line. Each of them weighs an integer number of pounds. For each rhinoceros (except the leftmost one), its weight plus twice the weight of its left neighbor ...
2
votes
1answer
215 views

Dancing, dancing, dancing

My friend Kim told me: Last week I went to an afternoon party. I was the last guest to arrive. I looked around and counted (not including myself) altogether 14 grown-ups and 17 children. ...
11
votes
2answers
873 views

Vladimir and his motorbike

Vladimir wants to take his two daughters to their Dacha (weekend house) which is located 66 kilometers away from their home. Vladimir's motorbike has a maximum speed of 50 kilometers per hour; with ...
8
votes
2answers
471 views

Bumblebees and Honey bees

The current market price of 175 honey bees is more than the price of 125 bumblebees, but less than the price of 126 bumblebees. Is it possible to buy 3 honey bees and 1 bumblebee for altogether at ...
5
votes
3answers
827 views

Witches and Cats

This being Halloween, we will consider the problem of witches and their cats. The whole crowd is out tonight, naturally, and an interested observer, seeing them fly by (yes, witches' cats fly), ...
19
votes
3answers
2k views

An arithmetic puzzle from 1645

The following problem is found in the first Norwegian arithmetic, published in 1645 by Tyge Hansøn: Three hundred oxen large and small, A cattle owner wanted to buy: 3 for 63 daler he got. ...