AMC 10 · 2024 · #18
Grade 10 geometry-2d
Pick an answer.
Tool #17 (Visualize Spatial Relationships) is the primary move because the whole problem is one mental rotation repeated. Lining up diagonals forces every card to share a single center, and every corner of every card is half a diagonal away from that center, so the entire growing stack lives on one circle. Once that is seen, a card is no longer a rectangle to be re-drawn — it is just four marks on a circle, and a placement just turns those four marks. Tool #1 (Draw a Diagram) supplies the coordinates that pin down the size of that turn, which comes from an angle hidden inside the rectangle. Tool #5 (Look for a Pattern) then takes over: the turn is the same every time, so the four marks step around the circle by a fixed angle and the picture must repeat. Tool #3 (Eliminate Possibilities) finishes it — there are only four corners that could land on B, so test all four and keep the earliest.
Every card shares one center
Sharing a diagonal means sharing a centre.
Lining up the diagonals nails every card to the same pin, so the only thing that ever changes is direction.
Lining up the diagonals nails every card to the same pin, so the only thing that changes is direction.
▸ Why?
All four corners sit the same distance from that pin, so a direction alone says where a corner is.
▸ Why?
Turning or flipping a card moves it without stretching, so those distances never change.
Find the hidden 30-60-90 triangle
The root three summons a 60 degree angle.
The √(3) in the side length is a signal that a half-equilateral triangle is hiding inside the rectangle.
8.G.B.7Draw A DiagramThe diagonal cuts off 15 degrees
An isosceles triangle delivers 15 degrees.
Two equal sides force two equal angles, which turns one leftover angle into the exact tilt of the diagonal.
10.G-CO.C.10Draw A DiagramRead the four corner directions
The corners sit at 75, 105, 255 and 285 degrees.
Since all corners are the same distance from the center, a direction alone says exactly where a corner is.
10.G-GPE.B.4Visualize Spatial RelationshipsOne placement turns things 30 degrees
Reflecting rotates the whole set by 30 degrees.
Flipping over a diagonal and turning by the angle between the diagonals do the same thing to the four corners.
8.G.A.1Visualize Spatial RelationshipsTrack the marks around the circle
Compute which placement lands each corner on B.
A fixed turn repeated turns a geometry question into counting steps around a 12-hour clock.
4.OA.C.5Look For A PatternTake the earliest hit
The earliest is the 6th.
Same direction plus same distance from the center means the very same point.
8.G.A.3Eliminate PossibilitiesLining up diagonals pins every card to the same center, so each new card is just a 30° turn — and any of the four corners, not only the one named B, is allowed to be the one that lands on B.
- Every card shares one center
- Find the hidden 30-60-90 triangle
- The diagonal cuts off 15 degrees
- Read the four corner directions
- One placement turns things 30 degrees
- Track the marks around the circle
- Take the earliest hit