NOTE: All submissions to any of these BTs receive an entry into a drawing for a free 2017 Calendar of your choice. Correct submissions receive at least one more.
1A. There are several interesting facts/traits surrounding the number 2017. Share one or more that you find. (NO ‘wrong’ answers here!)
1B. One special date in 2017 is 1/17/17. It is the only “Year-Product-Day” [YPD] this whole year! Can you define (or guess) what a YPD is? If so, when will the next one occur? When is the next year with only one YPD? When is the first year this century without a YPD? (Hint: We haven’t had it yet, if at all.)
2. Find a 5-digit number where the sum of all the digits is 25, and each digit is 2 more than the digit preceding it.
3. What number is 20 more than twice the number that is 10 more than 15 times one-half of ten?
4. Counterfeit Coin, Part 1 You have 12 coins in front of you, knowing that one of them is counterfeit. All you know is that the ‘bad’ coin is slightly heavier than the others. You also have a balance scale in front of you, and your task is to determine the ‘bad’ coin using only three (3) weighings*. How do you accomplish this?
*A weighing occurs any time you set something on both sides of the scale.
5. Miss Demeanor often goes the wrong way on a one-way street – RIGHT PAST a police station. How does she manage to do this without ever getting a ticket?
6. Find three consecutive integers that have the same product as sum.
7. Find a fraction which is equivalent to 1/2 and whose numerator and denominator have a product of 72.
8. Find the smallest positive integer which is divisible by each integer from 1 to 10 inclusive.
9. Fill in the missing letters: I V X __ ___ D M.
10. Consider a 7 x 7 x 7 stack of smaller cubes, all of the same size. How many of these are totally hidden from view?
11. You are facing south. You turn left, make an about-face, turn right, and then turn right again. Which direction is behind you?
12. (All names are street names.) Belvedere is parallel to St. Anthony. Davidson is perpendicular to River, which is parallel to St. Anthony. Is Davidson parallel or perpendicular to Belvedere?
BONUSES
All (legitimate) submissions to any of these Bonus BTs receive an entry into a drawing for the Rollin’ On The River book when it is released. All correct answers receive at least one more. (Numbers at the end of each BT represent maximum possible entries to be awarded.)
A. Counterfeit Coin – Part 2 Same situation as #4 above. Twelve coins, one is counterfeit, but you don’t know if it’s too light OR too heavy. You still have (only) 3 weighings to determine the bad coin. NOTE: It can be done, but it takes some clever thinking. (Solutions will need to explain carefully for full credit.) [3]
B. (See #1B above for details.) What is the only year (last two digits) each century that has seven (7) Year-Product Days? [2]
C. (See #8 above). Find the corresponding smallest integer divisible by each integer from 1 – 20. [2]
D. In #10 above, how many of the cubes have only 1 face visible? [2]
E. Here’s a fun BT I posted a year or so ago, and didn’t get too many nibbles. It’s not as hard as it might look, so I’ll pitch it out again. See 3LogiciansBar. [2]
BT 3 looks to be 190 and BT 2 would be 1,3,5,7,9 I think.
1A. 2017 is the 306th prime number
2. 13,579
3. 190
5. She is walking on
the street, not driving.
6. -3, -2, -1 AND
1, 2, 3
7. 6/12
8. 2520
4. Weigh 6 coins on each side of the scale – the side with the “bad” coin will weigh heavier. Take off the 6 from the lighter side, then weigh 3 coins on each side of the scale. Again, the side with the “bad” coin will be heavier. Take off all 3 coins from the lighter side. Lastly, choose 2 of the remaining coins and weigh them. If one side is heavier than the other, that is the “bad” coin. If the scale is balanced, the remaining coin that was not weighed is the “bad” coin.
9. L C (Roman numerals)
Bonus C. 232,792,560
(not sure this is the SMALLEST integer, but I know it is definitely divisible by numbers 1-20)
1. To count from 1 to 2017 it would take you 33 minutes. It is also a prime number.
2. YPD means that if you multiply the day and the month that it equals the year. It will occur next year in 2018 on 1-18-18. The one without a YPD I think is 2027 because it is a prime number also.
3. 190
4. You will weigh 6 on one side and 6 on the other. Take the coins that are the heaviest and weigh them doing 3 on one side and three on the other. Take the heaviest coins and weigh one on one side and one on the other. IF they are equal then the counterfeit coin is the one not being weighed. If one is heavier than the other then that is the counterfeit coin.
5. Miss Demeanor is walking not driving.
7. 6/12
9 L C
10. 150
Bonuses
D. 156
Answer to “counterfeit coin riddle” is found at https://www.youtube.com/watch?v=tE2dZLDJSjA on You Tube. Very clever but I can/’t take credit. Love the Krazy Pics.
11. West
12. Perpendicular
2. 25 = sum of digits of 5 digit number is 13579.
3. 210 = (((15 × 5) + 10) × 2) + 20
4. Counterfeit coin see below 5.
5. There are multiple possible answers, I think the most likely answer is she is walking. However, 1. she could be parallel parking and when backing up she is going the wrong way or 2. she could be a police officer on a bicycle going the wrong way or 3. she is a foreign diplomat with diplomatic immunity or 4. She is related to President Trump and no police officer dares to give her a ticket or 5. she is a police officer and likes to dare other officers to give her a ticket or 6. she is the Police Commissioner’s mother and no police officer will ticket her or 7.she is the head of a crime family and the last officer who gave her a ticket disappeared and she let everyone think she made him disappear.
4. Counterfeit coin:
#1. Weigh 6 coins on R scale and 6 coins L scale.
#2. Take off the coins that rise in first weighing, keep the 6 coins that drop in the first weighing. Select 4 coins of the kept coins weigh 2 on R scale and 2 on L scale. There are 2 possible results: A. One side drops or B. The scales balance.
#3. If 2A occurs take the 2 coins that drop, put one on R scale and the other on L scale, the coin that drops is counterfeit. If 2B occurs take the 2 kept coins that were not weighed in 2. and place one on R scale and the other on L scale , the coin that drops is counterfeit.
6. 3 digit number, sum = product. 123, 1+2+3=1×2×3=6
7.product of digits =72. The fraction is 6/12. Factors of 72 are 3,3,2,2,2. 6 x 12 = 2×2×2×3×3=72
8. Smallest whole positive number divisible integers 1-10. The answer is 1260. The number must be a multiple of 10, 9, 4, 5, and 7. The other integers, 2, 3, 6 are accounted for in 10, 9, 4. 1260 = 10×9×7×5×2.
9. Fill in the blank. LC. The letters are Roman Numerals in order of value.
10. 7×7×7 cube. There is an inner cube that us 5×5×5 that is not visible from the outside. However, it us not clear from the question if the bottom of the 7×7×7 cube is visible. There are 2 possible answers: 1. The bottom of the 7×7×7 cube is sitting on a glass floor or the cube can be picked up and is visible from below, the answer is 125. 2. The 7×7×7 cube is sitting on a wooden floor and cannot be picked up thus the bottom of the cube is not visible, the answer is 125 + 49 = 174.
11. Directions. South is behind. Turn left to East, about face to West, turn right to North, behind is South.
12. Streets. Answer us Perpendicular to Belvedier. Belvedier, St. Anthony, and River are all parallel, if Davidson is perpendicular to one it is perpendicular to all.
11. You are facing east, so WEST is behind you.
12. Davidson is perpendicular to Belvedere.