Snow problem
Raymond looked out of his window and was delighted to see the snow 5cm deep on all of the garden except the path, where the snow had already melted. The garden is rectangular, with the longer edge being 16 metres and the shorter edge being 10.5 metres. The path is another rectangle, 0.5 metres wide, and runs from the short side nearest the road to the front door.
Raymond started some rough calculations on a piece of paper. The snow could be collected up into a snowman which consisted of three spheres for the base, chest and head in ratio of volumes 3:2:1. In his (rough) calculations he uses 4x the radius cubed for the volume of a sphere and ignores the compression of the snow. Given this method, what radius should the base sphere of the snowman be?
Insulation calculation
Sven Svensen is nearing the final phase of his preparation for his trek across the arctic. An important part of his preparation is to put on weight to help protect him from the cold. Starting today, and for the following 20 days, he must gradually increase the number of calories he eats by 100 every day, starting from his usual 2,540 calorie daily intake. It doesn’t really matter what he eats to gain the weight, so he’d prefer to do it by eating Wazoo multi-nutrition bars, which contain 140 calories each. On how many of the days where he follows this plan will he be able to consume nothing but Wazoo bars?
Advent attempt
A competition runs in December where once each day every player can guess a number from 1 to 10,000. One lucky number for the month will win a prize and this number doesn’t change. Players can guess again the next day if they are wrong.
In one house, Steve is guessing the number each day. Lottie, his daughter, has been given a 24-door advent calendar, but she doesn’t understand numbers yet, just that she can open one door each day. After a third number had been guessed and a third door opened on 3 December, was it more likely that Lottie had opened the correct three doors in any order, or that Steve had picked the lucky number at some point in the past three days?
Pinned down
Janice is doing some urgent Christmas shopping. She has hit a snag as she has been asked for the four-digit PIN for her credit card and she has a terrible memory for figures. Fortunately, she has a flawless ability to remember mathematical procedures and can recall the correct method to find her PIN. She must begin by working out every possible three-digit number where each digit is different, they add up to eight and there is no zero. The sum of these three-digit numbers is equal to her four-digit PIN. She finds she can complete this operation quickly and without the need for a calculator, finishing her shopping just in time. Can you achieve the same result?
Wordplay
- Rearrange the letter groups to make three scientific instruments:TER TEL ABE OME AST OPE BAR ESC ROL
- Crack the code to find three scientists:ZULUXUS, JUMNYOL, YCHMNYCH
- Rearrange the letter groups to make three geometric shapes:GLE TRA REC IUM OID PEZ TAN IPS ELL
Puzzling presents
Zak has forgotten to attach name tags to his Christmas gifts. He remembers that the silver present wasn’t for his mum, and his dad’s present is either gold or red. The blue present belonged to either his sister or brother. If his mum’s present is red, his sister’s is gold, but if his mum’s is gold, his sister’s is blue. Who should receive each gift?
Test your metal
One chemist, one physicist and one biologist were asked to complete an important survey detailing their favourite metals. The results found that neither Ben nor the chemist prefer tin. The biologist, who is not Kim, does not like iron. Also, gold is the favourite metal of either Mary or Kim. What is each scientist’s favourite metal?
Answers:
Snow problem
Solution: One metre
Explanation: Begin by working out how much snow there is. The garden’s area is 10.5m x 16m, which is 168m2. The path is 0.5m wide and runs for the length (16m) of the garden, so its area is 8m2. This makes the area covered with snow 160m2. The snow is 5cm deep, so the volume is 160 x 0.05 = 8m3.
Now consider the volumes of the different parts of the snowman. The base is as large as the chest and head combined, as 1 + 2 = 3, so half the snow, 4m3, will be used for the base. Using the rough formula for the volume of the sphere, we can see that if 4r3 = 4m3 then the radius must be one metre and the diameter (width) two metres.
Insulation calculation
Solution: Three days
Explanation: First find something in the range covered by his plan that will definitely divide by 140. One such number is 2,940, which divides by 140 21 times. From this point, if we add or remove 100 calories we will only get to another number divisible by 140 every seven days, a total of 700 calories. It follows that 2,240, 2,940, 3,640, 4,340, 5,040 etc are divisible by 140. Starting at 2,540 and adding 100 for 20 days, we will reach 2,540 + 2,000 = 4,540 calories. It follows that 2,940, 3,640 and 4,340 are the only valid numbers in this range, so there are three days where he can eat nothing but Wazoo bars.
Advent attempt
Solution: It is more likely that Lottie has opened the correct three doors in any order.
Explanation: On day one Lottie had three doors she could open out of 24. On day two she had two out of 23 and on day three she had one out of 22. These events are consecutive, so we multiply them to find the probability, which is (3 x 2 x 1)/(24 x 23 x 22) or 6/1,2144. The odds of winning the lucky number prize are 3/10,000. 6/12,144 is a larger number than 3/10,000, so it is more likely that Lottie had opened the correct three doors.
Pinned down
Solution: Janice’s PIN is 3552.
Explanation: Start by finding the three-digit numbers. We cannot use any digit larger than a five so the only working combinations are (5,2,1) and (4,3,1). Six numbers can be made from the digits 521, with each digit in position one and the next two following in either order.
Considering these numbers, each position will hold each digit twice. This makes the sum for each position equal to (5 + 2 + 1) x 2 = 16. The same is true for the digits 431, so each digit will sum to 16 x 2 = 32. Consider 32 for each position:
3200 + 320 + 32 = 3552, which is Janice’s correct PIN.
Wordplay
- Telescope, barometer, astrolabe
- Faraday, Pasteur, Einstein (A becomes G, B becomes H, etc.)
- Ellipsoid, rectangle, trapezium
Puzzling presents
Solution: His mum gets gold, his dad gets red, his sister gets blue, his brother gets silver.
Test your metal
Solution: Ben - physicist - iron; Mary - biologist - tin; Kim - chemist - gold.
