This is a nifty problem from Presh Talwalkar.
“This is from a Manga called Q.E.D. I thank Sparky from the Philippines for the suggestion!
A string of beads is formed from 25 circles of the same size. The string passes through the center of each circle. The area enclosed by the string inside each circle is shaded in blue, and the remaining areas of the circles are shaded in orange. What is the value of the orange area minus the blue area? Calculate the area in terms of r, the radius of each circle.”
See the String of Beads Puzzle
Another puzzle by Presh Talwalkar.
“Thanks to John H. for the suggestion!
A square is inscribed in a quarter circle such that the outer vertices are on the arc of the quarter circle. If the quarter circle has a radius equal to 1, what is the area of the square?
I am told this was given to 7th grade students (ages 12-13), and I think it is a very challenging problem for that age group. In fact I think it is a good problem for any geometry student.”
See the Square in Quarter Circle
This is a nifty problem from Presh Talwakar.
“This is adapted from the 1994 Putnam, A2. Thanks to Nirman for the suggestion!
Let R be the region in the first quadrant bounded by the x-axis, the line y = x/2, and the ellipse x2/9 + y2 = 1. Let R‘ be the region in the first quadrant bounded by the y-axis, the line y = mx and the ellipse. Find the value of m such that R and R‘ have the same area.”
See the Putnam Ellipse Areas Problem
Here is another typical sum puzzle from Presh Talwalkar.
“Solve the following sums:
_____1/(1×3) + 1/(3×5) + 1/(5×7) + 1/(7×9) + 1/(9×11) =
_____1/(4×7) + 1/(7×10) + 1/(10×13) + 1/(13×16) =
_____1/(2×7) + 1/(7×12) + 1/(12×17) + … =”
The only reason I am including this puzzle is that Talwalkar gets very excited about deriving a formula that can solve sums of this type. This gives me an opportunity to discuss the “formula vs. procedure” way of doing math.
See the Incredible Trick Puzzle
This is a fairly straight-forward problem from Presh Talwalkar.
“A triangle is divided by 8 parallel lines that are equally spaced, as shown below. Starting from the top small triangle, color each alternate stripe in blue and color the remaining stripes in red. If the blue stripes have a total area of 145, what is the total area of the red stripes?”
See the Triangle Stripes Problem
This seemingly impossible problem from Presh Talwalkar turned out to be quite solvable upon reflection.
“A similar question was given to students in Thailand. For real numbers x, y, what is the minimum value of
√((x – 4)2 + (y – 10)2) + √((x – 44)2 + (y – 19)2)”
See the Square Root Minimum
This problem in logic from Presh Talwalkar recalled an article I wrote a while ago but did not publish. So I thought I would post it as part of the solution.
“Assume that both of the following sentences are true:
- Pinocchio always lies;
- Pinocchio says, “All my hats are green.”
We can conclude from these two sentences that:
- (A) Pinocchio has at least one hat.
- (B) Pinocchio has only one green hat.
- (C) Pinocchio has no hats.
- (D) Pinocchio has at least one green hat.
- (E) Pinocchio has no green hats.”
Actually, the question is which, none or more, of statements (A) – (E) follow from the two sentences?
See Pinocchio’s Hats
This is another infinite series from Presh Talwalkar, but with a twist.
“This problem is adapted from one given in an annual national math competition exam in France. Evaluate the infinite series:
1/2! + 2/3! + 3/4! + …”
The twist is that Talwalkar provides three solutions, illustrating three different techniques that I in fact have used before in series and sequence problems. But this time I actually found a simpler solution that avoids all these. You also need to remember what a factorial is: n! =n(n – 1)(n – 2)…3·2·1.
See a Nice Factorial Sum
This is another candle burning problem, presented by Presh Talwalkar.
“Two candles of equal heights but different thicknesses are lit. The first burns off in 8 hours and the second in 10 hours. How long after lighting, in hours, will the first candle be half the height of the second candle? The candles are lit simultaneously and each burns at a constant linear rate.”
See Two Candles
This is a nice brain tickling problem from Presh Talwalkar.
“A circle contains two tangent semicircles whose diameters are parallel chords. If the circle has an area equal to 1, what is the combined area of the two semicircles?”
See the Two Curious Semicircles