9 of 9 Items .... Source: Five Triangles

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An interesting Question ... is an algebraic method the only way to do this?



Prove that anywhere on Pascal's triangle, products of numbers in yellow and in orange are equal.

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Seems like a tough problem.




"Wait a minute. That would make it easier ..."

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Why did the puzzle creator only ask for circles A and B?

In the diagram, the 19 circles are to contain the integers 1 to 19 once each. The sum total along any of the lines is 38. Determine the integers in circles A and B.

Is it possible to find the rest of the circles, too?

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The floor lamp casts a shadow.


Straight line AB is drawn on a wide, spacious floor. A lamp is at a height of 8m above point C, which is located on line AB. Line CD is perpendicular to line AB. A rectangular solid 5m by 3m by 4m is on the floor as shown. The box is 2m from line AB and a meters from line CD. Assume the lamp is a point source and will cast a perfect shadow. Given that the rectangular solid casts a shadow with area \(90m^2\), determine the distance a (in meters).

• Does the 2m distance from the wall matter to the shape of the shadow?


• How does the distance a change the area of the shadow?


• What shape is that shadow on the floor?


• Are the edges of the shadow guaranteed to parallel with the edges of the box?


• How far away is that box from the light?

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What is this?

What can we do with it?

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I had asked "Is this really ok for a fifth-grader?

Five Triangles said, "We posed the question slightly differently, our diagram providing an important hint to bring it to a more manageable level for younger problem solvers."

Find \(x+y+z\)



What do you suppose the "important hint" was?

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When the integers from 1 to 30 are multiplied, determine how many consecutive digits starting from the ones (1s) position are zeros.

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100 students were surveyed about their preferences for bell peppers (capsicum), carrots and spinach, with the following results:

- 72 students said they like bell peppers, and 36 students said they like only bell peppers.

- 41 students said they like carrots, and 9 students said they like only carrots.

- 31 students said they like spinach, and 3 students said they like only spinach.

- 9 students said they like neither bell peppers, nor carrots, nor spinach.


What is the best way to model/analyze this survey data?
Graphically (and what type of graph)?
Tables? How?
Algebra?

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3
What are some ways to find this area?

Which way gives you the most satisfaction?