## Happy as L Solutions

In celebration of my 50th birthday, on Monday I published ten problems involving the number 50. In case you missed, here they are again:

1. There are 50 puppies to be adopted at a shelter, and 98% of them are hounds. How many hounds must be adopted so that 90% of the remaining puppies are hounds?
2. Let A = 1, B = 2, …, Z = 26. Find two common English words for which the product of the letters is 50.
3. What’s the least possible product of two prime numbers with a sum of 50?
4. While finding the sum of the numbers 1‎‑10, I got distracted and omitted some numbers. The sum of the remaining numbers was 50. How many different sets of numbers could I have omitted?
5. The square numbers are 1, 4, 9, 16, …, and the non-square numbers are 2, 3, 5, 6, 7, 8, 10, 11, and so on. What is the 50th non-square number?
6. Choose three numbers so that one number is selected in each row and each column. What’s the sum of the three numbers? 1. A two-player game is played on this number rack with five rows of 10 beads. One player chooses to be Odd, the other Even. The players take turns. On each turn, a player may slide one, two, or three beads from the middle to the side of the rack. Beads moved to the side cannot be moved again. When all beads have been moved, the Odd player earns one point for each row with an odd number of beads on each side, and the Even player earns one point for each row with an even number of beads on each side. The player with the most points wins. What is the optimal strategy, and who should win?
2. How many people must be present to have a probability of 50% that two of them will share a birthday?
3. Insert only addition and subtraction symbols to make the following equation true:

9    8    7    6    5    4    3    2    1 = 50

1. What’s the area of the square? (Inspiration from Catriona Agg, both for the puzzle and for the reduction in words.) As promised, here are the solutions.

1. Many people think you just need to adopt 4 hounds, since each hound represents 2% of the total. But that won’t work, because with each hound adopted, the total also decreases. Adopting 4 hounds would leave 45/46 ≈ 97.8%. To leave 90%, remove 40 hounds, which means that 9/10 = 90% of the remaining puppies will be hounds.
2. Since 50 = 2 &*times; 25, the letters B and Y can be used. One possible word is BY. Adding an A won’t change the product, since A = 1, so another common word is BAY. Since 25 = 5 × 5, another possible word is BEE. (An obscure alternative is ABY, an archaic word meaning to endure.)
3. 3 + 47 = 50, and 3 × 47 = 141.
4. The sum 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, which is 5 more than the result. Therefore, either 1 and 4, 2 and 3, or 5 by itself were omitted.
5. Through 49, there are 7 square numbers and 42 non‑square numbers. So 49 + 8 = 57 is the 50th non‑square number.
6. Doesn’t matter which three you choose, the sum will always be 50. Can you figure out how it works? And can you create a similar puzzle?
7. Sorry, you’re gonna have to figure out the strategy on your own…
8. This is the famous Birthday Problem, and 23 people are needed so the probability is at least 50%.
9. There are many correct equations. One is 9 + 8 + 7 – 6 + 5 – 4 + 32 – 1 = 50. (The trick is to notice that at least one pair of consecutive numbers should have no sign inserted between them, and those two numbers concatenate to form a larger number.)
10. The answer is 40,000 square units. You’ll need to prove to yourself that that’s correct.

Happy birthday to ME!

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• 1. Roger  |  March 17, 2021 at 2:16 pm

I enjoyed doing #10, and first solved it for the general case (‘n’ instead of 50). Then, for my grandson’s upcoming 9th birthday, I sent him the same puzzle featuring the number ‘9’. Instead of 50, I used a length of 9 inches, and asked him to work with his parents (both scientists) to solve it, expressing the answer in square feet. 🙂 And, yes, I concur: 40,000 is correct when n=50!

• 2. venneblock  |  March 17, 2021 at 2:34 pm

Sounds awesome, Roger! I hope he enjoys it. I’d be impressed by a nine-year-old who can solve that problem, for sure! Sounds like he may take after his grandfather.

• 3. Roger  |  March 17, 2021 at 5:49 pm

No, takes after his parents. He’ll need help from them but he’ll understand the concepts he’ll be introduced to – a fun problem. But the magic happens when it’s actually solved using 9″, after converting to square feet! 🙂