31

Question

A square whose sides each measure ten centimeters can completely fit inside of a regular hexagon whose sides each measure ten centimeters.
True
False

32
Six identical triangles can be formed by drawing two straight lines through an octagon's center point.
True
False

33
The number 64 is the next logical number in the following sequence of numbers: 2, 6, 14, 30...
True
False

34
Robert is taller than John. Charlie is taller than Robert. Therefore, John is the shortest of the three.
True
False

35
The sum of all the odd numbers from zero to 16 is an even number.
True
False

36
If each of seven persons in a group shakes hands with each of the other six persons, then a total of forty-two handshakes occurs.
True
False

37
Three congruent regular hexagons can be drawn in such a way that all of them overlap each other and create more than 6 distinct areas or compartments.
True
False

38
If a doughnut shaped house has two doors to the outside and three doors to the inner courtyard, then it's possible to end up back at your starting place by walking through all five doors of the house without ever walking through the same door twice.
True
False

Answer 1

Here are the answers with explanations for each statement:

31. True - A regular hexagon can inscribe a square whose diagonal is equal to the hexagon's side. In this case, a square with a diagonal of approximately 14.14 cm (which fits within the hexagon that has a side of 10 cm) can fit inside the hexagon.

32. True - Drawing two lines through the center of an octagon can indeed create six identical triangles.

33. False - The pattern in the sequence appears to be increasing with each term as follows: 2 (2), 6 (4), 14 (8), 30 (16), but the differences between terms are not constant. The next number would actually be 62, not 64.

34. True - If Robert is taller than John and Charlie is taller than Robert, it logically follows that John is the shortest of the three.

35. True - The sum of all odd numbers from 1 to 15 (i.e., 1, 3, 5, 7, 9, 11, 13, 15) equals 64, which is an even number.

36. False - If each of the 7 persons shakes hands with each of the other 6 persons, then the total number of handshakes can be calculated using the combination formula \(C(n, 2)\) where \(n\) is 7. This gives \(C(7, 2) = 21\) handshakes, not 42.

37. True - Three congruent regular hexagons can overlap and form more than 6 distinct areas or compartments.

38. True - It is possible to walk through all five doors without repeating, depending on the arrangement of doors. You can visualize it as a graph where each door is a vertex and each walk creates edges.

Summary of answers: 31. True 32. True 33. False 34. True 35. True 36. False 37. True 38. True