Student edition of Test 24
The original question order and point values are preserved.
Lesson 24.7 · Student exercise
Chapter 24 Exercise
Test 24: Classification. Organize cases so that none overlap and none are missing, then apply the same discipline to digit patterns, geometric figures, tilings, assemblies, and a balance-scale decision process.
Exercise directions
- Try the questions. Enter your answers and show your working where requested.
- Submit your attempt. Choose Submit exercise & open review when you are ready.
- Review and improve. Use the help available for this submission. Edit the original answer fields, then submit again.
Your work saves automatically in this browser. Use Print or Reset at the top of the page when needed.
Scoring notes
Conventions used on this page
Figures. Count a geometric figure once even when its vertices can be named in several orders. In Questions 1, 2, and 10, a side must lie completely on a drawn straight segment. In Questions 8 and 9, you may join the specified vertices with new straight sides. Do not measure the drawing; use the stated geometry.
Machines. The eight machines are identical, the three workshops are labeled, and every workshop receives at least one machine.
Repeated digits. “Exactly two digits are equal” means one repeated pair; the other two digits are different from each other and from that pair.
Selections. In Questions 7 and 12, order does not create a new choice. Each available number or segment may be selected at most once.
Fixed-board tilings. Dominoes are identical. Two coverings differ when at least one domino occupies a different pair of labeled cells.
Balance scale. Every comparison uses total weight, the heavier pan is kept, all weights are distinct positive integer grams, and no weight exceeds 15 grams.
Section I — Fill In
Questions 1–10 · 6 points each · 60 points
Triangles in a pentagram
6 pointsHow many different triangles are contained in this five-pointed star together with its outer pentagon boundary? Each triangle side must follow drawn straight segments; interior intersections may be vertices.
triangles
Diagram
Optional hint — try the question first
Give each triangle its three vertex labels. Naming the same vertices in a different order does not make another triangle.
One more step
Classify by the small regions inside the triangle, including the central pentagon where present.
Squares in an X-grid
6 pointsThe grid has 3 rows and 4 columns of equal square cells, with both diagonals drawn in each cell. How many different squares are contained in the drawn grid? Include both upright and tilted squares whose four sides lie on the drawn segments.
squares
3 × 4 cells, both diagonals in each cell
Optional hint — try the question first
Separate upright squares from tilted squares.
One more step
Within each orientation, fix a side length, then count the possible positions. All four sides must follow drawn segments.
Single round-robin tournament
6 pointsTen table-tennis players hold a single round-robin tournament: every pair of players meets exactly once. How many matches must be scheduled?
matches
Optional hint — try the question first
Give every match to the lower-numbered player.
One more step
Player 1 has nine new opponents. Player 2 has eight not already counted. Continue until the last new match.
Distribute identical machines
6 pointsA factory has three labeled workshops, A, B, and C. Eight identical new machines are distributed among them, and every workshop must receive at least one machine. How many different distributions are possible?
distributions
Optional hint — try the question first
Fix the number of machines received by workshop A.
One more step
For each A count, list positive B and C counts with the required remaining sum. Swapping B and C matters because workshops are labeled.
Tens digit greater than units digit
6 pointsAmong all two-digit positive integers, how many have a tens digit greater than the units digit?
numbers
Optional hint — try the question first
Fix the tens digit first.
One more step
If the tens digit is t, the ones digit can be any of 0 through t−1. Add these non-overlapping cases.
Four-digit numbers with one repeated pair
6 pointsThe four-digit numbers 1337, 1558, and 1371 share two properties: the thousands digit is 1, and exactly one pair of positions contains the same digit. How many four-digit numbers have both properties?
Exactly one pair. The other two digits must be different from each other and from the repeated digit. A triple repetition and two separate repeated pairs do not qualify.
numbers
Optional hint — try the question first
Separate the case where 1 repeats from the case where another digit repeats.
One more step
If 1 repeats, choose its other position. Otherwise choose the repeated digit and its two positions. In both cases keep the other digits different.
Choose three numbers with a divisible sum
6 pointsChoose three distinct numbers from 1,2,3,4,5,6,7,8,9 so that their sum is a multiple of 3. How many different selections are possible? The order of the three selected numbers does not matter.
selections
Optional hint — try the question first
Put the numbers into remainder classes 0, 1, and 2 modulo 3.
One more step
A sum divisible by 3 uses either three numbers from one class, or one from each class. Do not count the orders of a selection.
Lattice-point triangles of area 3
6 pointsEach small square has side length 1 centimetre. You may join any three non-collinear grid-intersection points, even when the triangle sides are not already drawn. How many different triangles with vertices at grid-intersection points have area 3 cm²?
triangles
3 × 2 unit-square grid
Optional hint — try the question first
Use coordinates for the twelve points. The triangle needs base × perpendicular height = 6.
One more step
You may draw sloping sides. Check every vertex triple systematically and record each set of three vertices only once.
Triangles sharing a regular-hexagon side
6 pointsChoose any three vertices of the regular hexagon and join them to form a triangle, drawing new sides as needed. How many such triangles share at least one complete side with the hexagon?
triangles
Regular hexagon
Optional hint — try the question first
Start with all choices of three hexagon vertices.
One more step
Which choices have no pair of adjacent vertices? Remove those instead of counting shared sides separately, which can double-count a triangle.
Triangles in a composite diagram
6 pointsHow many different triangles are contained in the figure? Count a triangle only when each of its three sides lies fully on a drawn straight segment.
triangles
Diagram
Optional hint — try the question first
Use the vertex labels and check all three sides of each proposed triangle.
One more step
AO stops at O. Do not extend it to the base. Group triangles by the smaller regions they contain.
Section II — Extended Response
Questions 11–13 · 20 points each · 60 points
Domino coverings of a fixed L-shaped board
20 pointsCover the shown board exactly with identical 1 × 2 rectangles. Dominoes may not overlap, leave a gap, or extend outside the board. How many different coverings are possible?
How to distinguish coverings. The cells are labeled only to identify positions. Two coverings are different when at least one domino covers a different pair of labeled cells. Rotating or reflecting an entire completed covering is not automatically treated as the same answer.
coverings
Fixed board and original cell labels
Optional hint — try the question first
Consider the joint above the bottom 2 × 2 block.
One more step
Either zero or two dominoes cross this joint; one crossing would leave an odd number of cells below it. Count the remaining rectangles in each case.
Choose segments to assemble a square
20 pointsThere is one rigid segment of each integer length from 1 through 9. Without cutting or bending any segment, select some and use every selected segment exactly once to assemble the four sides of a square. When two or more segments make one side, they may meet only at endpoints and may not overlap. How many different selections are possible?
Selection convention. A different method means a different selected set of segment lengths. Reordering the same selected segments along the sides does not create a new selection.
selections
Available segments
Optional hint — try the question first
Fix the square side length, then split the selected lengths into four groups with that same total.
One more step
The available lengths sum to 45, so the side length is at most floor(45 ÷ 4). List selected sets, not different arrangements of the same set.
When a three-weighing tournament selects the fifth-heaviest object
20 pointsEight objects have different positive integer weights, and no object weighs more than 15 grams. Xiaoping tries to find the heaviest object using the following three balance-scale comparisons:
- Divide the eight objects into two groups of four, compare the groups, and retain the heavier group.
- Divide that group of four into two pairs, compare the pairs, and retain the heavier pair.
- Compare the two remaining objects and retain the heavier object.
None of the three comparisons balances. The object retained at the end is actually the fifth-heaviest of the original eight objects. How much does that object weigh, and how much does the second-lightest object weigh?
g
g
Three-stage comparison tree
4 objects
vs
4 objects
2 retained
vs
2 retained
1 retained
vs
1 retained
At each stage, keep the heavier side. The final object is ranked fifth from heaviest, not first.
Optional hint — try the question first
Label the weights from heaviest to lightest and call the final winner x.
One more step
Its retained group must have exactly one object heavier than x. Compare the best pair containing x with the other pair, then combine that inequality with the first comparison.