Counting My Chickens Before They Hatch
I made a post yesterday about a particular word problem.
"3 hens lay 3 eggs in 3 days. How many eggs do 12 hens give in 12 days?"
I eventually looked at the problem from the point of view of a pattern recognition problem to see if the incorrect answer of 12 would ever make sense.
As I was falling asleep last night, that pattern was stuck in my head.
3, 3, 3, 12, 12, 48
I described it in my blog post as two sequences happening simultaneously.
One sequence is [...] taking the previous value and multiplying it by 4. But also, the number of times that value is repeated in the sequence is reduced by 1 each time.
This explanation didn't satisfy me. There's something else going on here.
I got out of bed and went to my whiteboard. The first thing I did was assign each place in the sequence a value.
a, b, c, x, y, z
- a and x are the number of hens
- b and y are the number of days
- c is the number of eggs we already know relative to the values of a and b
- z is the number of eggs we want to figure out relative to the values of x and y
There's a relationship here. If you plug in the example problem of 3, 3, 3, 12, 12, 48, the relationship soon emerges.
x / a = 4.
y / b = 4.
4 * 4 = 16.
16 * c = 48.
z / c = 16.
I tried different combinations of numbers, especially including values where x / a ≠ y / b. The relationship held.
Next, I generalized the relationship into an algebraic equation.
z = ( x / a ) * ( y / b ) * c
Because multiplication is commutative, the assigned values of a and b can be swapped so long as x and y are also swapped. But c and z must be the produced quantities.
Enter Leibniz
What the hell did I just discover? After a little research this morning, this appears to be an application of the product rule, originally discovered by Gottfried Leibniz in the late 1600s.
About those SAT questions
In my blog post from yesterday, I mentioned how much I hate those SAT pattern recognition questions because the human mind is capable of finding a pattern to fit almost any sequence. Does this formula constitute a pattern?
The variables in my equation have very specific meanings. Plugging in random numbers will always result in an answer, but it will be devoid of meaning.
We can turn this into a sequence, though. Every sixth number in the sequence is the result of plugging the previous five numbers into my formula. We can keep going on like that forever.
There's a precedent for this type of relationship: the Fibonacci sequence. It's a sequence. It's right there in the name. It's a much simpler sequence from mine since each number is calculated using only the previous two elements of the sequence.
Nonetheless, I have successfully defined a sequence. The problem with this is, if I'm allowed to introduce constants into my equation, given enough time I can deduce an equation that will fit any of the possible answers given to me in a multiple choice SAT question. All answers are equally valid. That means every question like this should be answered with the "not enough information" choice.
As much as I'd like to blow up this entire class of question, there is one major exception. My explanation only holds if the element you're being asked to identify is the next element in the sequence. If you are given elements of the sequence after the one you're being asked to identify, the number of equations that can describe the sequence is finite.
In case you were interested, here are the first 20 elements of the pattern. It's not too interesting since it almost immediately flies off towards infinity.
- 3
- 3
- 3
- 12
- 12
- 48
- 768
- 12288
- 3145728
- 51539607552
- 54043195528445952
- 928455029464035206174343168
- 66902235595591869424607154817945084517941248
- 20705239040371691362304267586831076357353326916511159665487572671397888
- 461742260113997803268895001173557974278278194575766953140019063331004552009732936852510932674432605165702671761408
- 12747311827602410368550116957259247055910234825448766156786659316809685436660453825739900054365502856053343907669916024368317904103664973745828479490398096446651950500049437256761475072
- 1961990857885010963650680631823733783986947287058472181598664041205476778182866435229245078397880599598867198945714990524634665440110438312253035444585920679431378376737835263443800281688666590920035555814156459042403919909802020427093300283095165328595800136295143219176104095151237375519507873792
- 8336703089455133366253152637284505809216106471025844872390916830078815575581716044809646733032619265778616285213946759452207573085365746121822190631552401897417984702241327995637007318439703819819237017739078217612263549262265907559878444210418396373430070436733784433880601967804216389727083671042268193066961537250359120881286712926652917288624603217799822741930541296725501513875590257340980893439367556323057845622863001578088832839595986733772796229913557054493030117443371008
- 21808713661883597881563482374816415506066458402606988584590146896407494641065003461831469007829236921156939848394548172685483468715249357409552864052535890566787091743536561319816497267475461380795954996794821697173460608140121128860067371567611284145986832882650074993192221238569222473848150948841889432872711728052735326108039592413636631969494641800703444401927343994874942236492255358796779598962031186861060517805812614909395833001503943326406130958029852004294995639392298626246110742915597698101176592399241095915330685964471084067208889434900400998183916750423928183756736534872617595679477941742688478064711541163297658079970897067805016422557754150686381214242124153995378971740082925111939907759963687990210368621967550245292511732835681608681678929932964441461096448
- 60604256854022455093416113615681329650788523232643268568446634166405803425760396292576876300156809099996745624104215236437174153009846337725481235827858011322629332419404707777286667575322924328823740197618709269757729497276279418795160516272276681232893064946547876708458499090359385417773135547670386025358232488397503658100102738605685493774721048809628900878842525522041634515650698010767584128105038647212056710796143039680022833147354811277939756240661313557159264926210709840939253717480829082139147488902509754199282883647950287165352778879846479794425781195146308500281075929994839691232702249051198079689462500764679392797712366748255171373161585181058901273664905793866974434550689497775178372280249086096851329412585713177961696929890483108780291025804376980672734520729475599923391166557111156608828732488219244373540917445750173201442703216860370344268672583211344894439092265361992781877161021646396073051125966332668484609638585655435717364152897091976636524200621392911128114208469486815282540111983881432019772211891335584199847624097880333496939806842751687644513206237638629944268587441328304542108028079325327137992235014007415763478728662561674659220731113476318565711033678672613928453409029313113871212397039223962680514829246978326528