# Puzzler

A broker instructs his assistant to select five equities for a client's portfolio and to make a note of the price of each. But the assistant gets confused. Instead of pricing the equities individually, the assistant makes as many pairings of the five equities as possible, and gives the broker the sums of the values of these pairings. The sums the assistant reports are: \$20, \$22, \$23, \$24, \$25, \$26,

A broker instructs his assistant to select five equities for a client's portfolio and to make a note of the price of each. But the assistant gets confused. Instead of pricing the equities individually, the assistant makes as many pairings of the five equities as possible, and gives the broker the sums of the values of these pairings. The sums the assistant reports are: \$20, \$22, \$23, \$24, \$25, \$26, \$27, \$28, \$30, \$31. Can you determine the values of the five individual equities?

Please email your solution to John Kador ([email protected]) using the subject line “Equity Pairs.” The deadline is March 15, 2009. Two accurate responses, based on originality and creativity, will be selected to receive a signed copy of John Kador's How to Ace the Brainteaser Job Interview. Good luck.

### BRAINTEASER #2: CIRCULAR REASONING

If three points are placed at random around the circumference of a circle, what is the probability that all three points will reside in some semi-circle, as in the diagram below?

### SOLUTION TO PREVIOUS PUZZLER: AVERAGE SALARY

To recap: A financial services firm has 10 employees. Although no one wants to divulge their individual salaries, they all agree it would be useful if they knew the average salary of the group. Devise a way for the group to determine the average salary of the group without anybody knowing the individual salaries of anybody else.

We received over 40 entries. A number of creative solutions were offered. Congratulations to our two winners who will receive a signed copy of How to Ace the Brainteaser Job Interview. Their solutions:

John Sergio, Chief Operating Officer at Maxim Group LLC in N.Y.: Employees are numbered one through 10. Employee One picks a random six- or seven-digit number. He/she emails this random number to Employee Two who is asked to add his/her salary number to it, and email that sum to Employee Three, who is asked to do the same and email that figure to Employee Four. This continues for all the employees through Employee 10 who then emails this sum back to Employee One. Employee One adds his salary and subtracts his random six- or seven-digit number and divides by 10. He is left with the average salary of all employees and it is derived with none the wiser about any particular employee's salary.

Roger J. Anderson, financial advisor with Bremer Investments/Invest Financial Corporation in South St. Paul, Minn.: Each employee fills up a coffee cup with a penny for each thousand dollars they make. One by one, each employee pours their pennies into a garbage bag. Then they count the pennies and divide by 10 to get an average. Using different coins to represent hundreds, tens and units of salary, this process can calculate the average salary to the desired degree of precision.

John Kador, the author of 10 books, published Charles Schwab: How One Company Beat Wall Street and Reinvented the Brokerage Industry. His website is www.jkador.com.

ANSWER TO #2: The probability is 75 percent. To see why, position Point One at the bottom (6 o'clock position) of a circle. Then imagine Point Two starting at the bottom and traveling clockwise around the circumference. With Point Two also at the 6 p.m. position, Point Three could be placed anywhere on the circle and the three points would have to be in a common semi-circle. As Point Two starts its clockwise journey, the options for Point Three shrink somewhat. If x denotes the number of degrees traveled by Point Two, then Point Three could be placed anywhere within an arc of 360-x degrees to ensure that the three points would all fit in a semi-circle. This pattern would continue until Point Two was at the 12 o'clock position (when technically, Point Three could again be placed anywhere), and then the pattern would reverse itself as Point Two traveled down the right side of the circle. You can graph it. For any value between 0 and 360, the lined area of the diagram indicates the possibilities for Point Two to satisfy the semi-circle condition. You will note that the lined area is precisely three-fourths (or 75 percent) the area of the square.