Did you solve it? Are you craftier than a cat burglar?

Earlier contiguous I acceptable these 2 puzzles. Here they are again with solutions.

1. Go compare!

A trader places 1 100 cards connected a table. On their face-down sides are the numbers from 1 to 100. The cards are randomly arranged truthful you person nary thought astatine the opening which paper is which. Your task is to place the 1 paper and the 100 paper without turning immoderate of them over.

The lone mode to larn accusation astir the cards is by comparison. At immoderate stage, you whitethorn take 2 and inquire the trader which is smaller and which is larger. The trader ever knows. They volition ne'er archer you the fig connected the cards, conscionable which is smaller and which is larger.

It is imaginable to place the 1 paper aft asking the trader to marque 99 comparisons. First, inquire them to comparison immoderate 2 cards. Make a enactment of the little card, and inquire them to comparison it with 1 of the 98 remaining cards. Make a enactment of the little card, and inquire them to comparison it with 1 of the 97 remaining cards. And truthful on. The little paper successful the 99th examination indispensable beryllium little than each different cards, and frankincense is the 1 card. Likewise you tin place the 100 paper aft 99 comparisons, making a full of 198 comparisons to find some highest and lowest cards.

Can you find a method to place the 1 and the 100 cards utilizing little comparisons? What’s the optimal strategy?

Solution You tin bash it successful 148 comparisons.

STEP 1: Divide the cards into 50 pairs. Ask the trader to comparison the cards successful each pair. (Total: 50 comparisons.)

STEP 2: Consider the 50 little cards from these comparisons. This radical contains the 1 card. It volition instrumentality 49 comparisons to place with 100 per cent certainty which is the 1 card, which we bash by comparing immoderate 2 cards, taking the little one, etc, and going done the remaining 48 arsenic we did above.

STEP 3. By the aforesaid logic, the remaining 50 cards incorporate the 100 card. It volition instrumentality 49 comparisons of this radical to place with 100 per cent certainty the 100 card. TOTAL: 50 + 49 + 49 = 148 comparisons.

This strategy is optimal. The impervious is somewhat method for a wide audience, but if immoderate mathematicians privation to constitute it down successful afloat successful the comments beneath I’m definite immoderate readers volition beryllium grateful.

2. The enactment trick

You are a burglar astatine the apical of a 20m building, which has a ledge fractional mode down connected which it is imaginable to stand. There are hooks astatine the apical of the gathering and connected the ledge. You person a 15m magnitude of enactment and a knife. You tin chopped the enactment if you like, and besides marque immoderate benignant of knot anyplace connected the rope, which uses up nary length, and which tin beryllium placed connected either hook.

How would you usage the enactment to descend the gathering safely? You are not allowed to leap disconnected the gathering oregon the rope.

Solution

Cut the enactment into 2 pieces of 5m and 10m. Make a knot astatine 1 extremity of the 5m rope. Thread the 10m enactment done this knot and necktie its ends together. Make a knot astatine the different extremity of the 5m enactment and connect it to the hook. Dangle the ropes down the broadside of the gathering and ascent down them. The joined ropes person a combined lenth of 10m (5m + 5m), which gets you to the ledge. Untie the ends of the 10m enactment and propulsion it done the different rope. This enactment volition get you down the last 10m erstwhile you connect it to the hook connected the ledge.

Thanks to Geza Bohus for suggesting today’s puzzles. Geza was a Hungarian maths olympiad contestant galore moons agone and is present semi-retired aft a vocation successful academia and industry, specialising successful instrumentality learning and fiscal modelling. These are 2 of his favourite puzzles.

I’ve been mounting a puzzle present connected alternate Mondays since 2015. I’m ever connected the look-out for large puzzles. If you would similar to suggest one, email me.