Last week’s delicious puzzle:
This past Saturday’s sneak-peek at spring let us break the rules a little and have dessert — ice cream, specifically — before dinner. Both Max and Nora got to enjoy the sugary treat, as shown below.
As the kids ordered their cones, I wondered just how many distinct combinations could be spooned up by our local ice creamery?
This would be, of course, subject to a few guidelines:
- Sugar cones can bear up to two scoops, waffle cones up to three.
- Fruit flavored scoops cannot be combined with other fruit-flavored scoops.
- If a cone has one chunk-based scoop in it, all scoops need to be chunk-based.
The following flavors are currently listed on the menu for selection:
Fruit flavors Chunk-based Other flavors Strawberry
Lemon Cello
Orange Sherbet
Watermelon Sherbet
Pineapple Coconut Chunk
Bananas Foster ChunkRocky Road Chunk
Pineapple Coconut Chunk
Chocolate Fudge Chunk
Peppermint Stick Chunk
Bananas Foster ChunkChocolate
Vanilla
Maple
CoffeeWith order counting (i.e., Chocolate atop Vanilla on a Sugar Cone is not the same as Vanilla atop Chocolate on the same cone), how many distinct cones can be created?
As with most of these combinatoric puzzles, there are generally two schools of thought: functional decomposition and brute force. I will walk through most of the solution by breaking it into smaller bits, but truth be told I solved the puzzle through brute force as I wrote it.
With 13 different flavors, there are some 26 different single scoop cone options available. There is no need to worry about conflicting flavors or required neighbors, as each scoop stands alone [+26].
Dual cones are where the challenges begin. Looking at the non-chunk fruit flavors (4 of them), they can be combined with both themselves and the non-fruit, non-chunk flavors (for of these) for a total of 20 combinations per cone type [+40]. Looking at the non-fruit chunk flavors (3 of them), they can be combined only with themselves or the fruit-chunk flavors (5 total) for 15 combinations per cone type [+30]. Both of the fruit-chunk flavors (2) can be combined wither with themselves or with a non-fruit chunk (4) for another 8 combinations per cone type [+16]. finally, each of the non-fruit, non-chunk flavors (4) can be combined either with another non-fruit, non-chunk flavor (4) or a single non-chunk fruit flavor (4) for 32 more combinations per cone type [+64].
So far, we have 26 different single scoop cones, and 150 different double-scoop cones for a total of 176 cone combinations.
Wheh!
The triple-scoop cones are similarly complex. Do recall that only the waffle cone can support three scoops, so the previous doubling to account for two cone types does not apply.
Let’s start with the non-chunk fruit flavors (4) again. Each can be combined with either itself twice [+4], with itself again with a non-fruit, non-chunk [+16], with a non-fruit-non-chunk and itself again [+16], or with two non-fruit, non-chunk flavors (4^2) for a total of 64 combinations [+64].
Starting with a fruit-chunk (of the 2), we need to stay in the chunk family. We can triple up on each flavor [+2], or double up on it an pick a different non-fruit chunk of the 3 [+12]. If we go with only one fruit-chunk on the cone, we need to pair it with some set of two non-fruit chunks (of the 3 available) [+18].
If we base our cone with one of the three non-fruit chunk flavors, we can go for all three non-fruit chunks [+27], two non-fruit chunk flavors and a single fruit-chunk [+36] or combine a single non-fruit chunk with a pair of fruit-chunks [+6].
Our last option is to start the cone with a non-fruit, non-chunk flavor (of the 4.) We can’t include any of the chunk flavors (fruit-chunk or non-fruit chunk), but we can mix them with either zero non-chunk fruit flavors (i.e., all non-fruit, non-chunks) [+64], a single non-chunk fruit flavor [+128] or two scoops of the same non-chunk fruit flavor [+16].
All totaled, there are a total of 409 different triple-scoop cones available, all in a waffle cone.
Overall, there are some 585 different distinct cone and cream combinations available for sale at our local creamery.
No wonder it takes Max and Nora so long to order! A complete list of all 585 cone and cream combinations are available upon request.
Congratulations to Blaine & Felicia. Their reasonably correct entry was chosen at random from those submitted this week. A fabulous $50 spending spree (gift certificate) from ThinkGeek is on its way!
Many thanks to everyone that posted a solution, and for reading GeekDad. Happy puzzling!
