Maths Puzzle: Cookie Jars

These puzzles are the second batch I'm taking from Math-E-Magic by Raymond Blum, Adam Hart-Davis, Bob Longe and Derrick Niedermann.  The first was a geometric question; these are based on algebra.
These puzzles are entitled Cookie Jar and Fleabags, but they are very similar to a wide range of puzzles (typically related to the relationships between people's ages).

Cookie Jar
Joe and Ken each held a cookie jar and had a look inside them to see how many cookies were left.  
Joe said, "If you gave me one of yours, we'd both have the same number of cookies."
Ken replied, "Yes, but you've eaten all of yours - you have none left!"
How many cookies does Ken have?

This is a relatively straightforward puzzle, helped by the fact that Joe has zero cookies, and there's only one other constraint - if Ken gives Joe a cookie, they'll have the same number (one).  So, if Joe will have one cookie after the transaction, then so will Ken.

But that isn't the answer.  We have to remember that Ken has one cookie after the transaction, but that he also had the one he would give to Joe - so he has two.

Fleabags

Two shaggy old dogs were walking down the street.
Captain sits down and says to Champ, "If one of your fleas jumped onto me, we'd have the same number."
Champ replies, "But if one of yours jumped onto me, I'd have five times as many as you!"
How many fleas are there on Champ?

This one is going to take a little more work - and we can use algebra to help solve it.

Let's have the number of fleas on Captain as A, and the number of fleas on Champ as H (taking the second letter of the two dogs' names).

If one flea jumps onto Captain, he will have A+1.  And if that flea has come from Champ, then he will have H-1.  And these numbers are the same, so A+1 = H-1  (1)

Now, if one flea jumps from Captain, he will have A-1.  And this number is five times greater than Champ's new total H+1.  So 5(A-1) = H+1    (2)

If A+1 = H-1 then A+2 = H (from 1)

And we can use this new value of H in (2), to give us 5(A-1) = (A+2) + 1

Expanding and simplifying:
5A - 5 = A + 3
4A = 8
A = 2

Captain has two fleas.

And since A+2 = H, Champ has four fleas.

Maths Puzzle: Arrange Nine Coins into Ten Straight Lines

This puzzle is taken from Math-E-Magic by Raymond Blum, Adam Hart-Davis, Bob Longe and Derrick Niedermann.  I've owned my copy of this book for a number of years and have referered to it in the past top help with problems I've been working on separately.  More recently, during a few idle moments (when there's not been enough time, energy or enthusiasm to do anything bigger) I've started solving some of the the puzzles it poses.  I've even (horror of horrors) started writing IN the book (but only in pencil).

Here's the first puzzle I looked at:

Nine Coins (page 29)

Wendy got into trouble in her math class.  She was sorting out money she planned to spend after school, and accidentally dropped nine coins onto the floor. The teacher was so upset that she told Wendy to stay at school until she could arrange the nine coins into at least six lines with three coins in each line.  Can you do it?  Wendy did, and in fact she arranged her nine coins into ten lines, with three in each line.  How?
The first question - can you get nine coins into six straight lines, is fairly straightforward, especially if you realise that nine is a square number.  If you arrange the nine coins into a 3x3 rectangle you can achieve six lines (three horizontal and three vertical).  

If you're more careful, you can arrange them into a symmetrical rectangle, or even a square, so that the diagonals form two extra lines, bringing the total to eight.  It's not ten, but it's getting us closer.

Working on the principle of increasing diagonal lines, in order to reach ten straight lines, we need to adapt the central row (or column) of coins so that they can provide the extra lines we need.  By moving the two coins at the end of the central row, we can achieve the addition more diagonal lines - see below...

The new diagonal lines are shown in the paler grey colour. We have lost the vertical lines at the edges of the square, but we have gained four diagonal lines, bring our total up from eight to ten.  The diagram below shows the solution with all ten lines shown:  three horizontal, one vertical, two long diagonals and four short diagonals.

One KPI too many

Three hypothetical car sales representatives are asked to focus on increasing their sales of hybrid cars for a month. They are a good cross-section of the whole sales team (which is almost 40 sales reps), and they each have their own approach. The sales advisor with the best sales figures for hybrid cars at the end of the month will receive a bonus, so there's a clear incentive to sell well.  At the end of the month, the sales representatives get together with management to compare their results and confirm the winner.

Albert

Albert made no real changes to his sales style, confident that his normal sales techniques would be enough to get him through top sales spot. 

Albert is, basically, our "control", which the others will be compared against. Albert is a fairly steady member of the team, and his performance is ideal for judging the performance of the other individuals.  Albert sold 100 cars, of which 20 were hybrids.

Britney 

Britney embraces change well, and when this incentive was introduced, she immediately made significant changes to her sales tactics.  Throughout the incentive period, she went to great lengths to highlight the features and benefits of the hybrid cars.  In some cases, she missed out on sales because she was pushing the hybrids so enthusiastically.

While she doesn't sell as many cars as Albert, she achieves 90 sales, of which 30 are hybrids.

Charles

Finally, Charles is the team's strongest salesman, and throughout the sales incentive month, he just sells more cars.  He does this by generally pushing, chasing and selling harder to all customers, using his experience and sales skills.  He doesn't really focus on selling the hybrids in particular.

Consequently, he achieves an enormous 145 sales, which includes 35 hybrid sales.   

Let's summarise, and add some more metrics and KPIs (because you can never have too many, apparently...).

	Albert	Britney	Charles
Total car sales	100	90	145
Hybrid car sales	20	30	35
% Hybrid	20%	33.3%	24.1%
Total revenue	$915,000	$911,700	$913,500
Revenue per car	$9,150	$10,130	$6,300

Who did best?

1. Albert achieved the highest revenue, but only sold 20% hybrid cars.
2. Britney achieved 33% hybrid sales, but only sold 90 cars in total.  She did, however, achieve the highest revenue per car (largely due to sales of the new, more expensive hybrids).
3. Charles sold 35 hybrids - the most- but only at a rate of 24.1%.  He also sold generally cheaper cars (he sold 110 non-hybrid cards, and many of them were either discounted or used cars)

So which Key Performance Indicator is actually Key?

This one is often a commercial decision, based on what's more important to the business targets. Is it the volume of hybrid cars, or the percentage of them? How far could Britney's drop in overall sales be accepted before it is detrimental to overall performance? And how far could Charles's increase in overall sales be overlooked?

Sometimes, your recommendation for implementing an optimisation recipe will run into a similar dilemma. In situations like these, it pays to know which KPI is actually Key! Is it conversion? Is it volumes of PDF downloads, or is it telephone calls, chat sessions, number of pages viewed per visit or is it revenue? And how much latitude is there in calling a winner? In some situations, you won't know until you suddenly realise that your considered recommendation is not getting the warm reception you expected (but you'll start to get a feel for the Key KPIs, even if they're never actually provided by your partners).

Image credits:
Albert:  https://satovconsultants.com/get-approach-gear-car-dealerships-not-ready-disruption/
Britney:  https://windsorstar.com/news/local-news/fcas-best-month-ever-as-overall-canadian-auto-sales-hit-new-record
Charles:  https://lifestyle.clickhole.com/beautiful-this-car-salesman-shaved-1-000-off-the-stic-1825120441 

Car: https://www.turbosquid.com/3d-models/3d-model-of-generic-hybrid-car-simple/942292