{"id":575,"date":"2024-12-17T08:00:00","date_gmt":"2024-12-17T07:00:00","guid":{"rendered":"https:\/\/control-advent.com\/?p=575"},"modified":"2025-11-28T19:06:17","modified_gmt":"2025-11-28T18:06:17","slug":"17-a-sweet-challenge-stability-analysis-of-santas-gingerbread-house","status":"publish","type":"post","link":"https:\/\/control-advent.com\/?p=575","title":{"rendered":"17 – A Sweet Challenge: Stability Analysis of Santa’s Gingerbread House"},"content":{"rendered":"\n

Authors of this question:<\/strong>\u00a0Helem S\u00e1nchez, Hafrsfjord Skole, Norway & Damiano Rotondo,\u00a0Institutt for data- og elektroteknologi (IDE),\u00a0<\/em>Universitetet i Stavanger, Norway <\/p>\n\n\n\n

Santa is building a huge gingerbread house at the North Pole, but he\u2019s worried it might not stay standing once it\u2019s covered in holiday decorations and delightful treats. The stability of this sweet creation depends on how many candy canes, k<\/em>, are used to support the roof. The clever elves have discovered that the relationship between the number of candy canes and the stability of the gingerbread house is somehow described by a polynomial.<\/p>\n\n\n\n

Inspired by Santa\u2019s jolly laughter, Rudolph the Red-Nosed Reindeer came up with a special method to check if the house will be stable which is now called the Rudolph-Hohowitz Criterion. This method involves creating a table where the first two columns are filled with the coefficients of the polynomial arranged in alternating order, while the remaining columns are filled using some magical formulas known only to the elves\u2014and to those who study the arcane art of control theory (hint: you might want to check out university courses near you if this sparks your interest!). If the first row of the table has all positive numbers, the house will be stable. But if any number in that row is negative, the house might collapse under all the decorations!<\/p>\n\n\n\n

This year, the polynomial that describes the relationship between the number of candy canes and the stability of the gingerbread house is:<\/p>\n\n\n\n

P(x)= x3<\/sup> + (1.5 – k) x2<\/sup> + (5 + k) x + 2<\/em><\/em><\/p>\n\n\n\n

The table looks like this, but the prankster elf Snoodle has erased one of the numbers! The other elves are looking at you to help fill it in:<\/p>\n\n\n\n

Column 1<\/td>Column 2<\/td>Column 3<\/td>Column 4<\/td><\/tr>
1<\/em><\/td>???<\/em><\/td>5 + k – 2\/(1.5 – k)<\/em><\/td>2<\/em><\/td><\/tr>
5+k<\/em><\/td>2<\/em><\/td>0<\/em><\/td>0<\/em><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Once the table is complete, what can you say about the stability of the house?<\/p>\n\n\n

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\"\"<\/figure>\n<\/div>\n\n\n

a. \ud83c\udf6d\ud83e\udd143\ufe0f\u20e3 The house is stable when an odd number of candy canes are used. That\u2019s odd, isn\u2019t it?<\/p>\n\n\n\n

b. \ud83c\udf6d\u2696\ufe0f4\ufe0f\u20e3 The house is stable when an even number of candy canes are used, which keeps the weight balanced.<\/p>\n\n\n\n

c. \ud83c\udf6d\ud83c\udfe05\ufe0f\u20e3 The house stays stable as long as more than four candy canes are used.<\/p>\n\n\n\n

d. \ud83d\ude2e\ud83c\udf8b1\ufe0f\u20e3 Surprisingly, the house is stable with just one candy cane, but adding more makes it collapse.<\/p>\n\n\n