Row $r$ of a number triangle has $2r-1$ entries. Odd numbers in order fill odd rows, and even numbers in order fill even rows, each sequence starting at $1$ or $2$. The first six rows start $1$; $2,4,6$; $3,5,7,9,11$; $8,10,\ldots,20$; $13,15,\ldots,29$; $22,24,\ldots,42$. Which row contains $2010$?
Only calculator keys $7,-,\times,\div,=$ work. Repeated $7$ makes $77$, etc. Pressing an operation immediately performs the preceding operation if any. Find key presses giving $34$.
Two parallel lines through opposite vertices divide the square into three equal-area parts as shown. Their perpendicular separation is $1$ cm. Find the square area.
Each floor of a building has ten apartments: $1$–$10$ on floor one, $11$–$20$ on floor two, etc. Mary\'s apartment number equals John\'s floor number. Their apartment numbers sum to $239$. Find John\'s apartment number.
Three couples spend whole dollars. Wives A, B, C total $2408$ dollars; A spends $400$ plus half of B, and C spends $204$ more than A. X spends four times his wife; Y spends $8$ more than his wife; Z spends $1.5$ times his wife. All six total $8040$ dollars. Find the couples.
Sixteen students A–P sit in four rooms of four over five days. No pair shares a room on more than one day. Replace each X in the smudged schedule shown with the correct student letter.
A $1\times4$ spaceship lands horizontally or vertically on a $7\times7$ grid. A mine in any occupied cell destroys it. Find the minimum number of mines guaranteeing destruction and show their placement.
Delete one number from $1$–$21$ and place the rest in a $4\times5$ table. All row sums must be equal and all column sums equal, possibly to a different value. Find all deletions and a table for each.
The six pieces shown consist of $2$ to $5$ congruent isosceles right triangles. Use $n$ distinct pieces, without overlap, to make a square. Find every possible $n$ and give a construction.
