Problem 16

What is the sum of the digits of the number 2¹⁰⁰⁰?

Problem 14

Which starting number, under one million, produces the longest Collatz sequence?

Problem 13

Work out the first ten digits of the sum of the following one-hundred 50-digit numbers.

Problem 12

What is the value of the first triangle number to have over five hundred divisors?

Problem 11

What is the greatest product of four adjacent numbers in the same direction (up, down, left, right, or diagonally) in the 20×20 grid?

Problem 10

Find the sum of all the primes below two million.

Problem 9

There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product abc.

Problem 8

Find the thirteen adjacent digits in the 1000-digit number that have the greatest product. What is the value of this product?

Problem 7

What is the 10001st prime number?

Problem 6

Find the difference between the sum of the squares of the first one hundred natural numbers and the square of the sum.

Problem 5

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

Problem 4

Find the largest palindrome made from the product of two 3-digit numbers.

Problem 3

What is the largest prime factor of the number 600851475143 ?

Problem 2

By considering the terms in the Fibonacci sequence whose values do not exceed four million, find the sum of the even-valued terms.

Problem 1

Find the sum of all the multiples of 3 or 5 below 1000.

El Calafate

I love the Perito Moreno Glacier. Growth from the center stretches out to the edge that ruptures crashing into the water. Even when the smallest white edge breaks the crack resounds and a raw blue of the glacier is exposed. And when you can pull your eyes from the face of the glacier to its body that snakes back into the mountains. The ice seems endless, and jagged till it turns soft in the distance blurring into the snow clouds. In this winter wonderland I was so tempted to stay, hike over the glacier, scale the peaks of nearby El Chaltan, but the promise of Buenos Aires pulled me north.  

Mountains and Lakes

Thanksgiving day I woke up in San Carlos de Bariloche, Angentina. The location is stunning. It sits amongst massive snow capped mountains with forested valleys that are claimed by lakes. friends from Valparaiso had recommended a hostel, and the owner Paolo from Buenos Aires spent an hour illustrating a map of where i should go for the best bike ride, hike and chocolate shop. This hostel serves fresh home made bread and jam in the mornings. I feasted with the snowboarding mohawked argentine and ponytailed german girls before leaving to catch the local bus to the famous Cerro Campinario.

 Gleaming with joy to be in a forest i leapt up the steep paths knobled with roots. The view from the top is majestic. I circled it several times before remembering my camera, and took insufficient photos. I descended slowly luxuriating in the vibrant yellow flowers, and cluttered trees.

 On the road to the bike shop a teapot shaped sign called my attention to a small cottage tucked into the mountain just over the lake, i promised myself a tea if i biked back before dark. By the time i arrived to the bike shop it was almost 3, and the shop closed at 6:30. The owners warned me that the loop usually takes 4 or 5 hours to complete. Tempted by the challenge i strapped on a helmet and radiantly neon green safety vest and pedaled off. At the first hill i disembarked legs weary from the cerro climbing, but soon i was in a rhythm. Fantastic songs snuck into my head and i bounced to the beat fighting to climb hills and  batteling winds across the exposed bridges. The entire ride was amazing, at times huge mountains cuddled the road from the left, or lake beaches on the right. Mostly the road just wove through the forest. Back at the shop i hinged off the bike trying to control my panting, they greeted me with confused smiles. Had i turned around or done the full circuit? The full circuit! I had returned in less than a hour and a half. We laughed and they invited me to their back room to freshen up. 

Knees wobbly i made my way to the tea shop.

Up the Volcano

The plan had always been to go from Santiago to Mendoza, cross into Argentina and bike around the vineyards tasting wine. My long contemplative walks led me question if this plan of winning was worth skipping over the temptingly close mountains and glaciers of Patagonia. At the terminal ticket counter i bought the ticket south, heart set on seeing a glacier. First an overnight bus to Pucon, the last place i would stay in Chile. I woke smiling to lush forests, mountians, a massive lake and the famous volcano. In the evening I took a kayak out into the lake till i found a small beach and filled a journal page in the black sand. Packing up to paddle back only when the sun began to set into the mountains glistening gold in the chops of the water. 

Before arriving i was decided not to summit the volcano, yet the next morning i  strapped into crampons and with an ice pick in hand began to to summit the smoking mountain.  Unlike the snow storming, altitude challenging climb, and leg shaking steep decent of Huayna Potosi, volcano Villarrica treated us with beach weather, and the decent was sledding on plastic saucers, tobogganing! Back at the hostel a small group of us dipped into the lake for a quick swim. And we all sat together in the sunny front lawn toasting the adventure with beers.

Santiago to Stay

Morning classes coaxed me into a routine. Wake for breakfast, take the metro to school, ask teachers for extra homework, take school work to a cafe or park, and celebrate with an adventure. Most adventures would really equate to a long walk, as i love to taste the tempo of each neighborhood. 

My favorite museum in Santiago was the Museum Memoria. I visited twice and could go back again. The comprehensice yet personal narration of the Pinochet dictatorship expanded my identity as a world citizen. 

My school organized activities for the students daily, my favorite was the soccer game. Thursdays after school a group of us met at the park and played on a small court. The games were fast, chaotic and fun. I ended up scoring 5 goals earning the title ‘estrella’. 

The Hearthrob Valpo

I loved Valparaiso from the first day. My taxi driver shared directions to  his favorite hidden gems, fish markets and cerros. The bright and beautiful Planeta Lindo is a hostel with an abuelita mural, roof top kitchen and terrance with spectacular views of the city and bay. The first afternoon i wandered the animated streets, admiring all the characters painted on the walls, stairwells and buildings. In the morning I attempted yoga poses on the sun roof and jogged to Pablo Nerudas house. I was fast friends with my Brazilian roommates and we took a field trip to Vina del mar to spend the day at the beach. The last day i went bouldering on some rocks by the sea. The grip of granite on my chalky hands was enough to counter the salty spray from the crashing waves. Monday morning i took a bus  at 5am, back to Santiago to start my first day of classes.