MouseHunt Math Quiz #5: Forcefield or Deathbot?

* PROBLEM

There is only one hunting location. You can buy cheese and traps by mail orders only. Goods are magically delivered within a few seconds after placing an order. But you can order goods only at midnight times.

You need cheese equivalent of 10 gold for hunting for a day (regardless of trap).
No minimum order value, but delivery charge of 200 gold per order will be applied, regardless of order value.

Forcefield cost = 500 gold
Deathbot cost = 550 gold
Drillbot cost = 1000 gold

You can use only one trap at a time, but you can switch traps whenver you wish.

The following is revenue for each trap.
Swiss army ~ 20 gold / day
Forcefield ~ 50 gold / day
Deathbot ~ 60 gold / day

Mice do not drop loot and do not steal gold.

In your inventory, you have:
a swiss army trap, 500 gold, but no cheese at all. Now it's midnight time, make a purchase decision now!

* EXAMPLE SCENARIO

You have 500 gold on day 0. Then you order cheese enough for 30 days (300 gold). Now you have no gold left (delivery charge = 200 gold).

On day 30, you have 600 gold (with a swiss army trap, you earn 20 gold per day). Then you spend all your gold for cheese enough for 40 days.

On day 70 (40 days later), you have 800 gold. Then you order a forcefield trap (500 gold) and cheese enough for 10 days (100 gold). Now you have no gold left.

On day 80 (10 days later), you have 500 gold (with a forcefield trap, you earn 50 gold per day).

* OBJECTIVE

Acquire digby drillbot as soon as possible.

* QUESTION

How many days do you need before you can purchase a drillbot?

***********************
* UPDATES

- Please start with: a swiss army trap, 500 gold, but no cheese at all. Please do not continue from the example scenario.
- Each and every order charges 200 gold for delivery, including an order of a trap without cheese
- Bonus question: assume that other things remain the same, but drillbot costs 800 gold (instead of 1000 gold).


***** ANSWER KEY *****

I use Wilbert Ho's format (Hope you don't mind :D)

MAIN QUESTION

Day 0 (START): Swiss/500 gold/No cheese
Buying 30 days' worth of cheese. (0 Gold left)
Day 30 (30d Later): Swiss/600 gold/No cheese
Buying 40 days' worth of cheese. (0 Gold left)
Day 70 (40d Later): Swiss/800 gold/No cheese
Buying the DEATHBOT (550g) and 5 days' worth of cheese. (0 Gold left)
Day 75 (5d Later): DeathBot/300 gold/No cheese
Buying 10 days' worth of cheese. (0 Gold left)
Day 85 (10d Later): DeathBot/600 gold/No cheese
Buying 16 days' worth of cheese. (240 Gold left)
Day 101 (16d Later): DeathBot/1200 gold/No cheese
Buying Digby Drillbot. (0 Gold left)

ANSWER = 101 days.


BONUS QUESTION

Day 0 (START): Swiss/500 gold/No cheese
Buying 30 days' worth of cheese. (0 Gold left)
Day 30 (30d Later): Swiss/600 gold/No cheese
Buying 38 days' worth of cheese. (20 Gold left) ***IT IS TRICKY RIGHT HERE***
Day 68 (38d Later): Swiss/760+20 gold/No cheese
Buying the FORCEFIELD (500g) and 8 days' worth of cheese. (0 Gold left)
Day 76 (8d Later): ForceField/400 gold/No cheese
Buying 20 days' worth of cheese. (0 Gold left)
Day 96 (20d Later): ForceField/1000 gold/No cheese
Buying Digby Drillbot. (0 Gold left)

ANSWER = 96 days.


***** EXPLANATION *****

There are two main decisions that you need to make.

(1) When/what to upgrade?

You can do a quick trial-and-error test and see that an upgrade is necessary for both main and bonus questions. So, an upgrade is necessary to acquire a drillbot sooner.

If you upgrade TOO EARLY (when you have little gold on hand), you will have to order several times and the delivery cost is very expensive. If you upgrade TOO LATE, you will lose an opportunity to harvest much higher earnings from Forcefield or Deathbot.

Timing is the toughest variable that you need to optimize. Did you realize that? lol...

(2) When/how much cheese to buy?

This is my recommended strategy: you should upgrade earlier if the number of orders remain the same.

For instance, you want to try Forcefield in the BONUS QUESTION. At the moment before upgrade, if you have 760-770 gold, you will need to order three more times; 780-890 gold -> order two more times; 900 gold -> you have enough gold to upgrade to FF and buy cheese that generates income of 1000 gold (drillbot 800 gold + delivery 200 gold) in one order. NOTE: If you have 750 gold or less, your progress will not go anywhere! Excel will help you determine these intervals quite quickly :).

In the example, you had better upgrade when you have 760 gold than 770 gold and 780 gold than 850 gold. This is because an early upgrade will give you higher profits from FF but will not cost you more delivery charges.

OK, now you should try 760, 780 and 900 gold for FF route. You need to work back a bit and calculate the day you have 1000 gold on hand.
760 gold -> hunt with Swiss Army for 66 days and earn 1000 gold on day 100.
780 gold -> hunt with Swiss Army for 68 days and earn 1000 gold on day 96.
900 gold -> hunt with Swiss Army for 110 days and earn 1000 gold on day 130.

Do the same thing for DB route:
800 gold -> hunt with Swiss Army for 70 days and earn 1000 gold on day 97.
820 gold -> hunt with Swiss Army for 92 days and earn 1000 gold on day 115.
920 gold -> hunt with Swiss Army for 102 days and earn 1000 gold on day 119.

Bear in mind that if you don't upgrade at all, you need 110 days. You see? 96 days is the optimum point!

That's the bonus question.

For the main question, many people just got lucky because the optimum route does not require you to carry gold until the end.
Here are numbers.
FF route:
760 gold -> hunt with Swiss Army for 66 days and earn 1200 gold on day 105.
790 gold -> hunt with Swiss Army for 69 days and earn 1200 gold on day 102.
940 gold -> hunt with Swiss Army for 104 days and earn 1200 gold on day 128.

DB route:
800 gold -> hunt with Swiss Army for 70 days and earn 1200 gold on day 101.
820 gold -> hunt with Swiss Army for 92 days and earn 1200 gold on day 119.
950 gold -> hunt with Swiss Army for 105 days and earn 1200 gold on day 125.

Bear in mind that if you don't upgrade at all, you need 130 days.


***** SOMETHING COOL *****

Here are some interesting facts:

Profit = Earning - cost of cheese
Trap cost breakeven = Trap cost / daily profit
Delivery cost breakeven = Delivery cost / daily profit

Swiss army
# profit = 10 gold/day
# delivery cost breakeven = 20 days

Forcefield
# profit = 40 gold/day
# trap cost breakeven = 12.5 days
# delivery cost breakeven = 5 days

Deathbot
# profit = 50 gold/day
# trap cost breakeven = 11 days
# delivery cost breakeven = 4 days


The delivery cost is the most challenging part in this quiz. Without it, this problem would have been a piece of cake. Now you know that under certain circumstances you should carry some gold before you upgrade, right? :)