Question 1
The product of the digits of 166 is $1\times6\times6=36$. List as many numbers as possible between 100 and 1000 whose digit product is 36.
IMSO 2008 — International Mathematics — Exploration
Question 2
Eight equal squares A, B, C, D, E, F, G, H overlap. The diagram is their top view. List the squares from top to bottom.
Question 3
Removing the upper-left cell of a $3\times3$ grid leaves 11 squares: eight unit squares and three $2\times2$ squares. Removing its central cell leaves only eight unit squares.
Shapes A and B each consist of six unit cells.
In each part, remove one copy of the specified shape from the $10\times10$ grid, keeping its boundary on grid lines:
(a) A, to leave the minimum number of squares.
(b) A, to leave the maximum number of squares.
(c) B, to leave the minimum number of squares.
(d) B, to leave the maximum number of squares.
Shapes A and B each consist of six unit cells.
In each part, remove one copy of the specified shape from the $10\times10$ grid, keeping its boundary on grid lines:
(a) A, to leave the minimum number of squares.
(b) A, to leave the maximum number of squares.
(c) B, to leave the minimum number of squares.
(d) B, to leave the maximum number of squares.
Question 4
Twelve blocks numbered 1 through 12 are given.
Arrange them into three stacks of equal height. The sums in all three stacks must be equal, and numbers must increase from top to bottom. The three examples below are considered the same arrangement. List as many different arrangements as possible.
Arrange them into three stacks of equal height. The sums in all three stacks must be equal, and numbers must increase from top to bottom. The three examples below are considered the same arrangement. List as many different arrangements as possible.
Question 5
Consider the $5\times5$ geoboard. The two illustrated crosses each surround five pins.
A. Find every square surrounding nine pins.
A closed shape can also use 24 pins: start at a pin, move to the next pin horizontally or vertically, visit each pin at most once, and return to the starting pin. Two examples are shown.
Some pins are always part of any such 24-pin shape; one is marked below.
B. Find all such compulsory pins.
A. Find every square surrounding nine pins.
A closed shape can also use 24 pins: start at a pin, move to the next pin horizontally or vertically, visit each pin at most once, and return to the starting pin. Two examples are shown.
Some pins are always part of any such 24-pin shape; one is marked below.
B. Find all such compulsory pins.
Question 6
Fill every cell of this $6\times6$ table with a number from 1 through 9, following all the row and column rules.
- R1: sum 21.
- R2: sum 39.
- R3: six consecutive numbers in order.
- R4: the largest and smallest entries are adjacent.
- R5: exactly two entries are 9.
- R6: sum 21.
- C1: exactly two 9s; the minimum entry is 2.
- C2: six consecutive numbers in order.
- C3: exactly two 3s; sum 21.
- C4: exactly two 8s; no entry is 4.
- C5: exactly two 6s; sum 40.
- C6: no entry is 7.
- R1: sum 21.
- R2: sum 39.
- R3: six consecutive numbers in order.
- R4: the largest and smallest entries are adjacent.
- R5: exactly two entries are 9.
- R6: sum 21.
- C1: exactly two 9s; the minimum entry is 2.
- C2: six consecutive numbers in order.
- C3: exactly two 3s; sum 21.
- C4: exactly two 8s; no entry is 4.
- C5: exactly two 6s; sum 40.
- C6: no entry is 7.