Registered Rep. presents the third installment of our brainteaser, The Puzzler (thanks, Car Talk).

Every other month (or so), The Puzzler offers a real-world puzzle and its solution. Many are real brainteasers used by corporations to test the mental agility of job applicants. This page also features a separate puzzle contest; the solution will be published the following month (for November and December's winners, turn to page 8). The most original response to the contest will receive a free copy of John Kador's book, How to Ace the Brainteaser Job Interview (McGraw-Hill, 2004).

February's Puzzler and Solution

A bizarre brokerage firm sells securities but only in very, very odd lots. Clients can order any securities they want, but the stocks are sold in bundles of three quantities. Clients can buy stocks in orders of 6 shares, orders of 9 shares or orders of 20 shares. So if you're a client of this brokerage and you wanted 45 shares of, say, IBM, you could buy five orders of 9 shares. If you wanted 50 shares, you could buy two orders of 6, two orders of 9 and one order of 20. But if you want 10 shares, you're out of luck.

Here's the question: What is the largest number of shares that you cannot order under this bizarre trading system?

Solution

There are a number of systematic ways to approach this puzzle, but the simplest approach is probably to start with 1 and work up, looking especially for prime numbers. You cannot buy 1 through 5 shares. You can obviously buy stocks in quantities of 6, 9 and 20 shares. You cannot buy 7, 8, 10, or 11. But you can buy 12. If you can buy 12, you can buy 18, 24 and all multiples of 6. If you can buy 18, you can buy 36, 72 and all multiples of 18. If you can buy 9, you can buy 45, 54 and all multiples of 9. And if you can buy 20, you can buy all the multiples of 100: 1,000, 10,000, 100,000, etc.

Working upwards, holes soon reveal themselves. You can't buy 31 shares, for example. You can't buy 43. But then something interesting happens. You can buy 44, 45, 46 and even 47 shares (three orders of 6, one order of 9 and one order of 20). Keep going for a few more numbers, and it appears that every number after 43, including primes such as 79 and 83, can be factored using 6, 9 and 20. My son wrote a program that confirmed this proposition for numbers up to 500,000. While this is not a definite proof, it's good enough for brokerage work. The largest number of shares that cannot be ordered under this scheme is 43.

Complete Puzzle Rules

  • The submission should include your name, address, affiliation and email address. No purchase required.

  • Entries must be received by March 15. One entry per person. Employees of Registered Rep. magazine and its affiliated publications are ineligible. Void where prohibited or restricted by law. Registered Rep. is not responsible for incomplete, lost, stolen, illegible, misdirected or late entries.

  • One winner will be selected on the basis of accuracy, originality and elegance, as judged by the magazine staff. In some cases, the winner will be selected in a random drawing from all eligible entries that contain a correct answer to the month's puzzle. The decision of the judges is final. We will notify the winner by email.

  • The winner will get his or her solution published in Registered Rep. The winner will also receive a copy of How to Ace the Brainteaser Job Interview inscribed by the author.

  • All entries become the property of Registered Rep. Each contest entrant consents to the publication of his or her name and puzzle answer, in any and all media and manner, now or hereafter known, in perpetuity without compensation. Registered Rep. is not responsible for any damages or losses relating to the puzzle or acceptance/use of any prize.

Writer's BIO: John Kador, the author of 10 books, published Charles Schwab: How One Company Beat Wall Street and Reinvented the Brokerage Industry in 2003. His Web site is www.jkador.com

February's Contest: Dividing Time

Challenge: On a standard 12-hour clock face, draw two straight lines that divide the clock face into sections. The numbers in each section must add up to equal sums.

Extra Credit

There is one “standard” solution to this puzzle. For extra credit, find an even more imaginative solution.

Please email your solution to john.churchill@penton.com, using the subject line, “February 2006 Puzzler” by March 15, 2006. Only one submission per reader, please. We will publish the solution we consider the most original. Good luck.