Welcome to The Riddler. Every week, I offer up a problem related to the things we hold dear around here — math, logic and probability. These problems, puzzles and riddles come from lots of top-notch puzzle folks around the world, including you, the readers. You’ll find this week’s puzzle below.
Mull it over on your commute, dissect it on your lunch break, and argue about it with your friends and lovers. When you’re ready, submit your answer using the form at the bottom. I’ll reveal the solution next week, and a correct submission (chosen at random) will earn a shoutout in this column. Important small print: To be eligible for the shoutout, I need to receive your correct answer before 11:59 p.m. EDT on Sunday — have a great weekend!
Before we get to the new puzzle, let’s return to last week’s. Congratulations to 👏 Sara Malec 👏 of Frederick, Maryland, our big winner. You can find a solution to the previous Riddler at the bottom of this post.
Now here’s this week’s Riddler, an interplanetary puzzle which comes to us from Robert Youngquist, a physicist at NASA’s Kennedy Space Center.
You are the CEO of a space transport company in the year 2080, and your chief scientist comes in to tell you that one of your space probes has detected an alien artifact at the Jupiter Solar Lagrangian (L2) point.
You want to be the first to get to it! But you know that the story will leak soon and you only have a short time to make critical decisions. With standard technology available to anyone with a few billion dollars, a manned rocket can be quickly assembled and arrive at the artifact in 1,600 days. But with some nonstandard items you can reduce that time and beat the competition. Your accountants tell you that they can get you an immediate line of credit of $1 billion.
You can buy:
- Big Russian engines. There are only three in the world and the Russians want $400 million for each of them. Buying one will reduce the trip time by 200 days. Buying two will allow you to split your payload and will save another 100 days.
- NASA ion engines. There are only eight of these $140 million large-scale engines in the world. Each will consume 5,000 kilograms of xenon during the trip. There are 30,000 kg of xenon available worldwide at a price of $2,000/kg, so 5,000 kg costs $10 million. Bottom line: For each $150 million fully fueled xenon engine you buy, you can take 50 days off of the trip.
- Light payloads. For $50 million each, you can send one of four return flight fuel tanks out ahead of the mission, using existing technology. Each time you do this, you lighten the main mission and reduce the arrival time by 25 days.
What’s your best strategy to get there first?
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And here’s the solution to last week’s Riddler, which asked you to complete this sequence: 10, 11, 12, 13, 14, 15, 16, 17, 21, 23, 30, 33, … Fifty-two percent of your submissions were correct. The missing numbers are 120 and 1,111. The sequence is the number 15 written in different number bases, or radices. Specifically, it’s the number “15” written in base 15, 14, 13, and so on down until base four. The missing numbers were 15 written in base three and base two.
The lot of us are familiar with base 10, in which integers have a ones place, a tens place, a hundreds place, and so on. (Each place is a power of 10.) But there’s no dictate from on high that numbers must be written in base 10. Binary numbers are base two (each place is a power of two) and are widely used by computers. Hexadecimal numbers are base 16 (each place is a power of 16) and are used to describe locations in computer memory and colors on web pages.
Now take, for example, base three (or ternary), which was the clue to one of the missing numbers in the sequence. Base three numbers have a ones place, a threes place, a nines place, and so on — all the places are powers of three. (Hence, base three!) So 15, when converted to base three, is 1x9 + 2x3 + 0x1, or 120.
Base two (or binary), has a ones place, a twos place, a fours place, an eights place, and so on. So 15 in base two is written as 1,111 (1x8 + 1x4 + 1x2 + 1x1 = 15).
(I suppose you could also go one further, and write 15 in base one: 111111111111111.)
Here’s a handy visualization of how the bases work in this sequence. The decimal number 15 is represented by the yellow entries:
From the Riddler home base: Happy weekend! You all complete me.